Mountain Bike for first tri?

It will be much more difficult.  I have certainly seen excellent mountain bikers, but a road bike (light, good position) will be faster and at the very least less tiring for you.

There are about a zillion threads on this topic around BT and maybe you can try a search for more evidence.  I always say, though, that I bought my first road bike on Craigslist for $75 and we put about $50 into it.  And you can never have too many bikes, imho.

Since you already know a road bike is superior, and you can certainly manage with a mountain bike, this is in direct response to whether you will find it frustrating.  You will.

Hey, Welcome to BT.  Do a forum search and just put in the words 'mountain bike.'  There are a ton of the same questions.  And, the same answers almost every time.  

YES, you should go ahead and use your mountain bike with slick tires.  Absolutely.  I've known plenty of people to do their first few triathlons on mountain bikes.  Get to know if you like triathlon before you 'plunk down over $1K.'  I was lucky enough to pick up a road bike from a garage sale for $15 that got me through my first 5 or 6 events before I decided I wanted a tri-bike.  

It's not a problem doing it on a mountain bike.  Sprints have lots of beginners, and there will be other mountain bikes there.  There's no doubt that a road bike or tri bike is faster, but the enjoyment of the race will come down to your training.  You're probably not out to win the race, so it doesn't matter if your bike is slower.  If the course is hilly, that will exaggerate the speed difference of the road bike vs. the mountain bike.

In my first year, I did my first two races on a mountain bike for pretty much the same reason you said, and had a great time!  It was especially fun passing road bikes!  There also happened to be a mountain bike division, so I was racing against those bikes for placement. 

And as for a comparison, I did do one of those Sprint races again my 2nd year on a tri bike.  My bike time for 12 1/2 miles was roughly 4 minutes faster on the tri-bike vs. the mountain bike.  The speed difference was something like 19.5 mph vs. 17.5, and this was a hilly course.

easier/harder?

depends on your perspective on easier/harder.

It is harder to go the same speed for the same amount of energy on a MTB vs. a road bike.

It won't be any harder because you only can exert x amount of energy, and that depends on what kind of shape you are in.  lets jsut assume that translates to 15 mph on a MTB and 20 mph and a road bike.  Of course it is going to be harder on the MTB to go 20 mph, but you cannot look at it that way as it's apples and oranges.  If you want to palce in a race, it's not going to happen on a MTB.

I rode a MTB in 3 training races last eyar, and it didn't discourage me from coming back.  All it did wan encourage me to get a tri bike.

I didn't even swap out the tires and had a blast--I could not wait until the next one.

Most tri's have a Fat-tire division so you can race against those doing the same.  I did FT for my first year, 2 sprints and an Oly, then bought a used RB and then a year later finally bought a TB as I was hooked.

My biggest increase in speed came from using a combination of tires with a slick center bead and blocks on the side for some trail riding and increasing my pedal rate from ~ 70 to more like 90.  With a couple of months to go and some dedication to training you'll be fine.  Get a speedo with a candence function, ~$30-35, Cateye has a good wired unit.