You'll probably pull a muscle while grappling or get a bruise during stand-up (striking) if your partner overdoes it. They're somewhat common, but they're not supposed to happen. Pre-fight preparation and conditioning is supposed to build you up, not leave you walking into the ring feeling sore. About 3-4 weeks before you'll probably be going all out and sweating yourself out on the mats/flooring, but in the last 1-2 weeks before they event you'll want to take it down a notch, unless fighting in a fatigued state is cool with you, which in the sport, really isn't.
Overloading muscles is common practice for conditioning training, but most coaches advocate safe training (thus avoiding unnecessary setbacks) with practices like joint mobility exercises for warm-up and active recovery, and post-training stretching to keep everything from locking up and going sore the next day. This is the standard in the sport, from Martin Rooney to Ross Enamait. Mark Hatmaker was also a big advocate of safe training. It's smart. While the chances of you getting injured in HIIT are slim, you may feel a little sore after pumping iron on the barbells (no benches, please). That's normal though, and doesn't really count as an injury.
So there. Pulled (and sometimes overly sore and fatigued) muscles from working your grapple-game or working with heavier loads for conditioning, or the occasional bruise, sore or even cut from your striking game. Part of the sport, and nothing really serious. Training for combat sport over the years has taken in a lot of science, so you get dividends on your preparation up to the fight and are more than capable of bringing the pain when you're actually there - in 100% top shape, ideally.
Qualifications for my post:
Sambo/MMA Fighter
Former Judoka