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Onward trip: PIT -> PHX -> SLC (AA)
Return trip: SLC -> JFK/NYC -> PIT. (Delta)
Can I just skip my NYC to PIT flight as I plan to be in NYC for the holidays? I don't care about being refunded for that component of the journey. No checked-in luggage.

How would TSA/Airlines react?

Also I am staying in the USA on a student visa (unaware how that impacts this case, but my ID proof for TSA is always my passport).

Most terms of carriage state some variation of the theme:

If any portion of the ticket or leg of a flight is not used, any subsequent legs will automatically become invalid.

Furthermore, any checked luggage will usually be tagged for the final destination. (However, you can even get around this if you ask check-in staff nicely and they have a good day.)

However, you only propose skipping the very last leg of your return trip. As long as the airline doesn’t think you are using that excessively to get around its pricing schemes (cf ‘hidden city ticketing’), nobody will stop you and you will likely face no consequences.

Your final leg will be recorded as no-show. Since you do not have any checked luggage, that makes it all much simpler for the airline staff; they will wait maybe a minute of courtesy before declaring you didn’t show up. People do that all the time and airlines have complicated schemes of overbooking flights to a certain percentage because they know a certain number of people will not show up.

Other answers have mentioned that there may be repercussions if you do that too often and too regularly with the same airline. I have no reason to disbelieve the general validity of that statement. I do want to point out that you will have to do it a lot and very obnoxiously for them to actually decide that they care.

TSA has nothing to do with the entire process; it is all between the airline and you.

If you did this a lot, it's possible an airline might ban you from flying it, as this is called hidden city ticketing - fares are based on city pairs, not routes, so it's possible e.g. to fly NRT-YVR-SEA (Tokyo Narita-Vancouver-Seattle/Tacoma) for less money than NRT-YVR. Do it occasionally? Probably not a problem. Do it regularly? You may be invited to fly on another airline permanently.

You mention having no checked bags, but for the benefit of others, checked bags will go to the final destination.

Another warning point: if your flight gets cancelled or rescheduled, you may not end up passing through the city you want, as you booked passage from A to C, even though you bought A-B-C. The airline could fly you A-C directly, or A-D-C. (In my example above, you could get rebooked NRT-YYZ-SEA - Narita-Toronto Pearson-Seattle/Tacoma - and the airline would be within its rights.)

This is not allowed and could lead to you losing your frequent flier miles or being banned from flying that airline. That said, you could probably get away with it once.