I’m new to dming and a player wants their tiefling to not have a tail. I think it’s probably ok but I just want to make sure it doesn’t break the game or anything.
In general: You're the DM...the rules and setting details exist to enable you and your friends to have a good time playing the game together. If there are things that you don't like, you are free to change them.
To this specifically, there are no mechanical features of the Tiefling tied to their tail--it's purely cosmetic. So there will be no change to gameplay on account of a player deciding their Tiefling doesn't have a tail. Additionally, there is a sourcebook that sets a precedence for Tieflings looking different from others.
Your tiefling might not look like other tieflings. Rather than having the physical characteristics described in the Player’s Handbook, choose 1d4 + 1 of the following features: small horns; fangs or sharp teeth; a forked tongue; catlike eyes; six fingers on each hand; goat-like legs; cloven hoofs; a forked tail; leathery or scaly skin; red or dark blue skin; cast no shadow or reflection; exude a smell of brimstone.
From the Sword Coast Adventurer's Guide
So...you're only changing cosmetic details (not mechanics) and there is a precedent for Tieflings looking different from the baseline described in the PHB. So...go for it. There's no reason a Tiefling has to have a tail.