I ran DIII track for two years. While the commitment is going to be variable between schools (and even within the team by event type), I'm gonna disagree with the poster above because I don't know of any schools where you just need to "pop your head in every now and again" - that sounds like a club sport to me. Even during off-season when we were training "on our own" (I didn't run XC) we were still expected to show up to student led practices every day (or make up the equivalent actually on our own). The difference between DI and DIII track is more likely to be the level of competition than the time involved (and maybe a bit more leeway from coaches for academic issues and conflicts).
I really loved running track, so if it's something you have the time for it might be worth it. However, the reason I quit after two years was because I just wasn't able to commit the time - I wasn't trying to study for the MCAT, but I had a lot of other ECs on my plate that I wanted to devote more time to. Track can really eat up a lot of your free time very quickly. For me, the time commitment was generally 1-3 hour practices 5 days a week, then either competition (sometimes including substantial travel) or longer practice on the weekend, with one rest/active recovery day. This doesn't include the stuff we were expected to do on our own time - active recovery, lifting, rehab/prehab, informal team events/dinners, etc. it adds up really quickly. For reference, this was as a sprinter/hurdler - the distance folks had a somewhat different schedule (more morning runs, less lifting) but I'd imagine that probably took up more time if anything.
If you're planning to still do competitive irish dance, you may have a hard time fitting it all in - even if you technically have enough hours in the day, I know that I found it hard to balance track with similar performance related activities, so you may find yourself having to choose one over the other.
With all that said, I know several people who ran track for all 4 years of undergrad and are now in med school. It's absolutely doable, and if you'll enjoy it I say go for it. I really did enjoy my two years.