(edit: Oddthinking found an explicit denial that the pizza box had anything to do with it, so I accordingly have edited this answer.)
Report says "police monitored social media"
I believe the initial "pizza box" story came from this Twitter account which cites a scoop by the Romanian language newspaper Gândul. Here is a Google translation of their article:
Sources close to the investigation stated to Gândul that shortly after the completion of the computer analysis, the authorities waited for the right moment to catch the Tate brothers, who were always out of the country.
After seeing, including on social networks, that they were together in Romania, the DIICOT prosecutors mobilized the special troops of the Gendarmerie and descended, by force, on their villa in Pipera, but also on other addresses.
Emphasis added, for my analysis.
Claim is making two false inferences
People seem to be inferring from the article that the "social media" posts which prosecutors used to decide their timing included the pizza box video. This was questionable (now known to be false), because the article specifically mentions that they needed to know that both brothers were together in Romania, something which cannot be confirmed by the pizza box video.
The tweet makes a second incorrect inference which many others have repeated as well:
she made him so angry he inadvertently tipped off Romanian authorities of his presence in Romania
The article makes it clear that they relied on multiple sources of information to establish Tate's presence. They already knew he was in the country and, given the time that paperwork takes, they may have already been in the midst of writing a warrant at the time that he posted the pizza box video.
The claim that the video "tipped off Romanian authorities" is false, because they needed to locate both brothers. The claim that the video assisted authorities is explicitly denied by police:
Speculation swirled overnight Thursday that Romanian authorities were able to locate Tate after he posted a video in response to Thunberg containing a pizza box from a local spot that gave away his location. Bolla denied that this played a role in the detention or its timing. “It was a hard job gathering all the evidence” in the months-long investigation, Bolla said. (WaPo)