Any increase in population will put an increased strain on services and infrastructure. If a service is managed privately for profit then these additional people are customers and the business that serves expands to meet new demand which includes more jobs and more money in the local economy. If the service is publicly owned or not run for profit, then the cost of expanding to meet the increased demand falls to donors, taxpayers, supporting businesses, etc. So it's true that immigration, legal or otherwise, as well as births strain existing services, whether or not that strain is a concern or a benefit depends on specifics.
Specific to the United States. The numbers you get depends on who you get them from. According to President Trump, the cost of illegal immigration is in the billions of dollars.
We are not even into February and the cost of illegal immigration so
far this year is $18,959,495,168. Cost Friday was $603,331,392. There
are at least 25,772,342 illegal aliens, not the 11,000,000 that have
been reported for years, in our Country. So ridiculous! DHS
The source he used to get that number is unclear. Neither he or DHS has since verified the information.
The number of illegal immigrants is hard to pin down as well. A 2018 Yale study estimated the number of people illegal living in the United states is 22 million, but that study has faced a lot of criticism as well. https://journals.plos.org/plosone/article/comment?id=10.1371/annotation/b53601d7-9b42-450a-b23c-39f4d889fcbd
Getting to actual cost is also hard. For example, 1 more person riding a bus doesn't actually increase cost as long as the bus isn't full. 1 extra person going to the emergency room poses more of an immediate concern because they don't share service with other patients, but they also don't use the service every day. This study from the Commonwealth Fund says the US spends about $4200 of public money per person on healthcare in the united states: https://www.commonwealthfund.org/publications/issue-briefs/2015/oct/us-health-care-global-perspective, but those dollars aren't allocated per patient.
Also relating to cost, there's a false idea going around that illegal immigrants don't pay taxes but use public services for free. In fact, the opposite is true. The IRS gets 4.4 millions returns filed by people without social security numbers which includes illegal immigrants who file taxes as a means of creating a personal paper trail because they believe it will help them avoid deportation if they are caught. The revenue from all taxes paid by people without SSNs in 2015 was $23.6 billion dollars which doesn't count the revenue from sales tax which everyone pays at the register.
I can try to do math with these wacky numbers, but it won't mean much. If 4.4m people pay 23.6b dollars in taxes each pays $5400 in taxes. If $4200 is spent on healthcare per person then it looks like a net gain of $1200. However, the 4 million that pay taxes are far less than the 22 million that Yale's researchers estimate live here and still than the more common 11 million estimate. Using the latter, each taxpaying illegal immigrant would have to pay $10,500 in income tax just to cover the healthcare costs for themselves and other illegal immigrants who don't pay which means the US is at about $5100 loss for every 11 immigrants which is about $460 per person which goes up if you consider more than just healthcare and down if you consider more than just income tax, and that's only using the 4.4 million taxpayers figure which is certainly not comprised only of undocumented immigrants. <- see? meaningless.
Moving from cost to overall impact, there's evidence that communities with more immigrants have lower crime rates. The Journal of Ethnicity in Criminal Justice published this report in 2016 that shows that crimes rates actually fall the rate of foreign-born persons rise in a community, and communities which are comprised more of immigrants than nationals compared to other cities experience less violent crime per capita https://www.tandfonline.com/doi/abs/10.1080/15377938.2016.1261057. Less crime leads to less cost for police and courts and less violence leads to fewer emergency room visits. So there are arguments to be made that immigration lessens strain on some services, but it would be too big of a stretch to assume this completely offsets the other costs such as schooling, transportation and income assistance.
So how much does illegal immigration impact social services in the US? It's impossible to tell.