No. The date you enter your programs onto the NRMP rank list where it's Feb 1 2021 or March 2 2021 has no influence on the match/your chances. That said, it only takes a few minutes to fill it out though so I would go ahead and do it now to ensure you are:
1. Registered for the NRMP 2021 Match (Basically paid the the $80 or now $135 given the late fee).
2. Verified via your medical school. (this is mainly for non-US grads, US schools will verify everyone all at once).
3. Able to list all the programs you intend to match. Make sure you have the exact NRMP code for the program you are trying to match and make sure to include tracks as a match. I would definitely do all this with at least a business day to spare in the event you run into a glitch and need to call tech support.
When I did it, I actually officially certified way ahead of time (early Feb). It gave me the assurance that I was done unless I wanted to change things. You get an email and all and it's official. If you have to change your order, you can always edit your list by re-entering your password, changing things, and recertifying.