First, note that nearly all US doctoral students are funded as TAs with smaller numbers for the others. Very few bring external money. Research universities have a huge need for TAs to help with the undergraduate program which is part of their overall mission.
Second, every US university is different and even within a given university policies and procedures may differ by department, so an absolute answer is impossible.
But, I doubt that it would have any effect on admission, unless there are some special circumstances.
The question is (most places) primarily a way for the administration to get a handle on the funding requirements for the coming year. They would like to know whether you would be likely to accept an unfunded offer or would reject it out of hand. Checking yes is unlikely to decrease your chances, provided you are otherwise qualified.
I don't know the details of your fellowship but if it won't cover you for the entire time of your study then you will probably need funding, though not at this time. If it covers a couple of years then you can probably work out a TA (or other) later. Technically, however, you don't need the TA as a condition for acceptance so it might be a mistake to check the "yes" box.
Checking no, on the other hand, might possibly increase your chances by a bit, but only if you would be admitted anyway and only if the university needed to limit admissions for financial reasons. It won't push you ahead of any other candidate, though.
Note that the department probably has a limit (not too hard) on the number of "seats" available, and another limit (harder) on the funds available. All bets are off as those limits approach. At the margin, someone might think "We can't fund anyone else, but we have a few open seats. Is there anyone here we should still admit without funding." That can happen, but I doubt (weasel word) that it is very common.
When I was a grad student (days of King Arthur) we had about 150 doctoral students in the department but only a couple of them weren't TAs or RAs. Almost all TAs.
I don't much like giving answers with a lot of weasel words (likely, doubt, probably...) but things vary. Sometimes all you can get are things to think about since there are so many options and variables in US university education.
