Yes, it can. That doesn’t mean it will but it is a good possibility. The first three days are the most crucial to healing the tattoo. The rest of the healing time, about two weeks, is spent keeping the skin from drying out and being overly wet. The sweat can cause some of your ink to actually wash out of your skin, especially within the first three days. The biggest problem with washing out ink is that you will, more than likely, need a touchup. It’s not a big deal but you’ll have to wait until the tattoo is healed before you can have it touched up.
@Afos22 and @judochop the ink enters and sits in the upper layers of the dermis whereas the sweat glands are found in the lower layers. Here is a diagram showing where the ink is injected compared to the rest of the anatomy of the skin. Now, the sweat does rise to the top so it’s not like the tattoo is going to flush out like a waterfall. If you lose ink, it’s just going to dull the colors and make lines look spotty. It can all be fixed with a quick touchup. Just be careful because a touchup needs time to heal just like the original tattoo does and some artists/shops will only do a few (usually one) free touchups.
Another problem with sweat is the ease of spreading infection. Bacteria is all the time landing on you and a warm, sweaty area is a perfect breeding ground for bacteria, fungi and many other microbes that can cause plenty of problems for you and your tattoo. Those breeding microbes then get into the fresh wound and can wreak havoc on the new skin. If you have no choice but to sweat on it, make sure you are able to wash with ANTI-BACTERIAL SOAP frequently throughout the day and reapply whatever ointment you decide to use. (I REALLY recommend Aquaphor. The link will lead you to a site where you can read about it if you’d like. It allows the skin to breath but keeps the area moist for a longer period of time and with all the different cremes and ointments I’ve tried, it’s been head and shoulders above all others. It’s safe and extremely effective)
Like @judochop said, stretching can cause more serious problems. The movement issues cause more of problem with different locations for the tattoo. For example, if you were to get something in the ditch of you’re arm, (the bend opposite of your elbow) it’s more likely to develop problems because of bending issues than something like the upper arm, (bicep/tricep area) for obvious reasons. Depending on the location of your tattoo, this may or may not be an issue.
I would really suggest waiting until you have at least a three day stretch off work before you get your tattoo just to be cautious and safe. Developing problems with tattoos can make horrid experiences for you, your artist and often, your physician. It’s better to be safe than sorry. If you just can’t wait, then clean it more often than you actually need to. Always clean it before you reapply any ointment.
I hope this gave you somewhat of an idea of what to look for and what to expect. Your artist should be able to answer these and any more questions you can think of, as well as help you out in any way possible if a problem does develop. Good luck with your new ink!