There are dozens of articles out there like this one which states that Trump ordered the use of tear gas to disperse peaceful protesters so he could do a photoshoot. However I recently read that reporter Neal Augenstein (whom I am otherwise unfamiliar with) send out a series of tweets that read:
First on WTOP: U.S. Park Police is explaining its decision-making in clearing Lafayette Square, at least 20 minutes before DC's 7p curfew, on night 4 of protests. A source says tear gas was never used -- instead smoke cannisters were deployed, which don't have an uncomfortable irritant in them.
And, the source says Park Police didn't know President Trump would be walking across the park several minutes later. Park Police say the reason the crowd was disbursed [sic] with smoke cannisters is that at that moment, officers were being pelted with water bottles.
Another factor was that protesters had climbed on top of the structure at the north end of Lafayette Square that had been burned the day before.
Obviously, this is Park Police's side of the story. I'll be checking with @wtop reporters who were on the scene last night, to plug-in what they observed with what Park Police are telling me. Park Police will be releasing a statement later today.
Obviously folks in the White House will be asked today about the President's walk across Lafayette Square to St. John's Episcopal Church. But, my Park Police source says the agency made its decision to use smoke cannisters at that moment because of what was being thrown at officers, not because President Trump planned to make an unannounced walk to the church.
Park Police will be releasing a statement later today. We will keep asking questions. In theory it's possible another agency used tear gas, in addition to Park Police using smoke cannisters, but my source was on the scene, got a dose of smoke, but didn't feel the irritants of tear gas. Clearly, the phrase "tear gas" has been used widely in the reporting and I'm continuing to try to learn who used what. Will keep you in the loop.
I think there are three issues here: 1) Was 'tear gas' used? 2) Who ordered the attack? 3) Why did they order it?
Was 'tear gas' used?
According to the Parks Police smoke canisters and pepper balls were used (see also here) (more specifically, OC gas1 was used). At least the CDC considers those among the umbrella term 'tear gas' (while the Park Police does not).
Who ordered it?
I don't think anyone claims that Trump personally ordered the attack on protesters.
Instead it has been reported that it was ordered by William Barr:
Two federal law enforcement officials told the newspaper that Barr and other officials made the decision late Sunday or early Monday to extend the perimeter around Lafayette Square [...]
Barr found the perimeter had not yet been extended Monday afternoon and ordered law enforcement officials to complete the extension
Why did they order it?
The internal motivations of people are not really on-topic here. But the claim by the Park Police that the attack was a response to bricks and 'caustic liquids' being thrown is disputed by multiple media outlets who witnessed the event, by the mayor and district attorney, and by the fact that the decision to clear the park was made beforehand.
 The link is geo-restricted. This is a tweet by the author of the article. The full article is archived as well.
A Justice Department spokeswoman said Monday that Trump had directed Barr to personally “lead” the response to the unrest.(via the Washington Post)
Barr then directed the police to widen the perimeter around the White House.
...Barr was surprised Monday evening that the perimeter around the White House hadn’t been extended by an additional block, as was decided the day before; he [Barr] then told police to get it done immediately. (via New York Mag)
Were protestors cleared using Tear gas? Yes.
Claims to the contrary are based on semantics.
At least one spent canister recovered from the streets outside the park Monday by reporters was clearly labeled “Skat Shell OC.”
The OC stands for oleoresin capsicum, an oily substance derived from chile peppers that is often used in topical ointments and “heat” creams for arthritis relief and muscle pain. When it gets into the eyes, noses and lungs, however, it triggers searing, debilitating pain, coughs, sneezes and mucus secretion.
As police across the country use such chemical agents to quell protests and rioting, they also are triggering those major vectors for the spread of coronavirus.
Like the pepper-ball projectiles, the OC gas detected Monday is very much a lachrymatory product designed to produce tears. It clears crowds by making the air sting and burn — one reason the Park Police and other federal forces wore gas masks as they cleared a path for Trump to walk across the street and pose for photos outside a vandalized church.
“Acute exposure to smoke would not cause the severity of what I was seeing,” she said. “We’ve all been sitting near a bonfire, and you’re not walking away crying in agony because you can’t open your eyes.”
According to the Centers for Disease Control and Prevention, riot-control agents are “chemical compounds that temporarily make people unable to function by causing irritation to the eyes, mouth, throat, lungs, and skin.”
Several compounds fall under this category, according to the CDC.
Among others, they include chloroacetophenone (CN), more commonly referred to as “mace,” or pepper sprays. Such compounds are all typically referred to as “tear gas” because their most prominent effect is to irritate mucus membranes, including the eyes, which secrete tears as a protective response.(via the Washington Post)
edit: U.S. Park Police finally stopped lying and admitted they used tear gas:
Were the protestors cleared in order to do a photoshoot? Yes.
On Monday, in a moment combining the aspiring authoritarianism and chaotic bumbling that has defined the Trump presidency, federal law enforcement officials forcibly removed a group of peaceful protesters from a park near the White House so President Trump could stand in front of a church and brandish a Bible. (via New York Mag)
The administration disputes that dispersing the protestors was for the photoshoot, but remember that this administration disputes a lot of basic facts. (via the Washington Post)
On Tuesday, the administration offered conflicting explanations for the forcible removal of the protesters, seeking to separate the move from Trump’s visit to the church.
The White House asserted that the crowd was dispersed to help enforce the city’s 7 p.m. curfew, although District police had not requested such assistance. The Park Police said that its officers responded after protesters began throwing projectiles. (via the Washington Post)
TRUMP: You have everybody on tape, you gotta arrest all those people, you gotta try them. And if they get five years or ten years, they have to get five years or ten years. There's no retribution. So I say that and the word is dominate. If you don't dominate your city and your state, they're gonna walk away with you. And we're doing it in Washington, in DC, we're going to do something that people haven't seen before. But we're going to have total domination. And then you have to put them in jail and you have to authorize whatever it is, whoever it is you authorize, and with that I'll let Bill Barr say a few words and then I'll let Gen. Milley speak. Let's go, Bill. (via CNN)
TRUMP: If you're weak and don't dominate the streets they're gonna stay away from you until you finally do it, and you don't want that. Philadelphia, you better toughen up because what's going on in Philadelphia like New York is terrible. It's terrible. You better toughen -- they never leave. I know you want to say all...hundred people. You've got a big National Guard out there that's ready to come in a time like now. I tell you, what they did in Minneapolis was incredible. They went in and dominated. And it happened immediately.
All of the sudden my wife said to me, 'wow look at all these people.' They got out and were wearing dark black uniforms. They got out and they were there in the houses and they walked right down the street, knocking them out with tear gas, tear gas. Those guys, they were running. And the next night it was much less and then the next night, it's like you know what happened, they went to other cities. They're all looking for weak spots. Now what they're going to do is they're going to search out for perhaps smaller cities, smaller places. You've got to arrest these people. You've got to arrest these people and you've got to judge them, and you can't do the deal where they get one week in jail. These are terrorists, these are terrorists, they're looking to do bad things to our country. (via CNN)
TL;DR: All evidence points to yes. Ultimately, Trump was upset about his press coverage; ordered Barr to clear the protestors; and shortly after walked through the cleared area in order to do a photoshoot.
Regarding Trump ordering tear gas specifically:
Trump repeatedly talked about the need to "dominate the streets"; said "we're doing it in Washington"; praised the use of tear gas against protestors in Minneapolis; called American protesters "terrorists"; remarked that "you have to authorize whatever it is, whoever it is you authorize"; and directed Barr to personally "lead" the response.
I'll let the reader draw their own conclusion from that available information.