None if done right. The entire reason for twisted pair cabling and differential signaling is to get a clean signal through a noisy environment.
A bad install is going to result in an inconsistent low performance link. The good news is it isn't rocket science so just take your time and go slow. Yes power lines can add noise and you want to cross at 90 degrees at that reduces the effective noise (after the effect of the differential signalling) but power lines aren't the only thing to watch out for.
Don't use too much force when pulling the cable. If it snags then fix that at the source and resist the urge to pull with all your might. The reason is that when you overstress the cable it can cause the wire pairs to deform so they are no longer properly twisted and that reduces the EMI protection. These are the worst type of problems because visibly the cable will look exactly the same as a good one. It also will reduce the crosstalk between wires in the cable. Each wire pair has a different twist rate so they block signals from each other but when you high force the lowest twist rate wires tighten up first and can become in sync with its neighbors. Short answer is just take your time, do it right the first time and the wire will probably outlive you. Use excessive force to just get it done and you may end up with a error prone link that makes your wonder why you just didn't go wifi.
Other things to watch out for. If you use solid wire (you should it is better than stranded for in-wall installs) make you the wire can't move. That means no dangling ends. No putting a RJ-45 jack on one end and have it hang out of the wall. Solid core is much less forgiving than stranded and the wires will eventually snap. So staple it down periodically and have it hard terminate into jacks or patch panels.
Be sure to make a loop of cable (usually I do a loop with 2" radius twice. That gives you about 1' of slack in case the ends need to be reterminated. Make sure to secure both ends of the loop so it doesn't move. Remember terminations may need to be replaced and the cable will be cut short each time (if it gets too short someone may decide to pull on it to try and get some slack). Lastly take your time to ensure the cable isn't damaged. A kink in TP cable can kill the EMI resistance. When stapling the wire in place be sure not to crush the cable. Use properly sized staples. They should prevent the wire from moving easily without crushing down on the cable. It also isn't that unusual for someone rushing to put a staple right through a cable (even pros do it) so take your time.
Also I would make a extra run to each drop. Even if you leave it unconnected it can be used as a backup or if you needs change (maybe you will run video over it in the future). It doesn't really add any extra cost or time to make an extra run at the same time.
If you are truly worried there is STP (shielded twisted pair) cable but it requires special terminations, generally is a pain to work with, and honestly would be overkill for your situation. If it makes you feel any better, data and electricity cross paths all the time in commercial buildings even in datacenters and STP is almost never used.