Triathlons on a mountain bike

Just pedal harder.

Slicks will probably make the biggest difference. I did that when I started out before upgrading to a road bike.

I don't know about your MTB, but some have adjustable fork suspension and some have suspension locks.  It might be worth a look.  I suspect the less bouncy the better for flat surfaces.

For a little more elaborated answer, it depends on what you have now and what you really want to do. I have a couple sets of tires for my mountain bike. One for more hardpack or MUP use and another with deeper, knobbier treads for more messy conditions. I'd just go with the former and pedal hard. Actual slicks would be faster still, but it's still a mountain bike and I'd rather see how much I liked this before spending it. Maybe save it for a more fitting bike or just save it in general. At least until I was more sure of wanting to continue with this. If you find a good deal though, knock yourself out.

The locking shocks was a good suggestion too. My bike is 10 years old and the front does lock (Specialized Stumpjumper). I'd think this would be an even more common feature if the bike is any newer. Also maybe drop off any other extra stuff you don't really need for the race.

I know you said you didn't have the budget for a used bike, but in your situation I'd be looking for a vintage roadie for cheap. They can be found in good working condition for $50-100 and will be way faster and more comfortable than a MTB, even with slicks. I have an old Miyata that I got for $70 and have been on long rides on it and would not hesitate to do an OLY or even a half on it. Plus you get a bike that wouldn't be a drag to train on the road with. 

To me that's clearly money better spent than on slicks. And the old roadie will always be worth close to what you spent for it-- oftentimes more. And they are very cool. 

Exactly, and you don't have to mess with changing tires every time you go from dirt to road. 

I have a friend that started in triathlons on a vintage roadie and he was competitive in his age group. They were designed to go long and fast--they were the race bikes of their time. 

Just for fun and out of curiosity, I browsed Craigslist and found several examples on the first page that I'd do a triathlon on, all $100 or less. Good luck and have fun!