I'd replace the tires as you don't want to have a tire fail on a tandem, esp the front.
Without knowing the rim, it's hard to judge, but a wide rim with a narrow tire will make it easy to get pinch flats.
If you buy wider than regular road tires (i.e. bigger than 23c) make sure you get the wider tubes too. 19-21c size tubes in 25-32c tires make for easier flats too (not sure why but after perhaps 15-20 attempts at saving weight by doing so I stopped).
I haven't tried 25c tires on our tandem. It has 28c and it's fine so we haven't bothered with it. The two of us are probably 320-330 lbs, bike is about 40 loaded up. Personally I'm ready to relace the wheels with much narrower rims and experiment with 25s, but we have too much going on to worry about that.
Raising seatpost should be fine as long as you have enough post in the frame to avoid the post or seat tube bending. Difficult to say without pictures. You may have to fiddle with the captain's stem too. Your position should be similar to that of your single bike.
hope this helps,
cdr