Generally, the inside width of a rim is about 5mm less than the outside width.
While putting a 25mm tire on a rim with 3/4" (19mm) inside width might work OK, my recommendation is to go just a bit wider. I've had trouble in the past putting on tires that are barely wider than the rim. Like Mad Scientist says, the performance differences are not that big. I run Ritchey Tom Slicks 26x1.4 (~35mm) on my 17mm rims.
If you really want to be competitive in the triathlon, you need to get yourself into a more aero position, either by adding aerobars to your bike (if you can get that to work) or getting a whole new bike made for road riding. The biggest reason MTBs are slower than road bikes is that they put you in a more upright position causing more wind resistance. This is much less of a factor when riding off-road since speeds are slower.