Will Slick Tires Work for a Mountain Bike Ridden by a Heavier Rider? I've owned a Trek 3500 bike (shown below) for 2.5 years now and I like it very much. 



My bike has the same specs as under the link. It has the XL frame size and 26" wheels with Bontrager AT-550 36-hole rims. I've started looking into changing the Bontrager tires to slick road tires though as I've begun to ride more for exercise and not just recreational. 95% of my riding is on paved roads/trails. The problem is I have no idea what tires can work on my bike for me. I'm considering the high pressure Kenda K193 tires. Can anyone advise? I'm looking for the best tires I can get for around $35 each. Is it even possible to replace the 2" tires with 1.5" without needing to buy new rims and everything?

I'm riding ~35 miles on weekends and let's just say I'm right now a heavier guy (~300 lbs). I've ridden on the same tires almost daily since I first bought it, so I feel it's about time to change them anyway.
Those tires should work just fine. Any of the MTB slick tires listed on that site would work well with your bike and current rims. 

You can get too small a tire on too wide a rim, but it takes a much bigger jump than from 2" to 1.5 or so. There is a very conservative sizing guide on Sheldon Brown's bike pages. 

Given your weight, I would try and find the largest size slick you can. Something like the Schwable Big Apple in a 2.0" format1 supports a maxim load of 275 lbs per wheel, but it's outside your price range, or from that page the Kenda K1029 Kwick Roller Sport 26 Inch Tire (50-559).

The last number is key to understanding tire sizes. You have 559mm rims, so just about any 559 tire will work with your bike. The 50 is the nominal width of the inflated tire in mm's. ( i.e. 50mm ~ 2 inches ).



1 Schwalbe has a sizing guide too.