We recently handed on our Burley to a cousin. It received a lot of use in the time, since No. 1 girl was about 8 weeks old, in a car capsule strapped into the trailer.
It's important that the kids are comfortable. When the girls were younger, we used a folded up thermarest, lightly inflated, to give them a cushier ride, a better view and more of an incline for sleeping. Also a cushion to rest their head on. We never used helmets in the trailer, but always on bike seats or backs of tandems.
We often put in another kid in there to allow them a break from a bike seat if they needed a better sleep on a longer ride. Sometimes 2 kids didn't get on too well in the cramped quarters. We didn't do too many ride with our 2 girls together, as the older one was able to ride on the back of the tandem when we started with No 2. Hopefully your 2 would get on well. The older style KoolStops where you can have the kids face different directions allows for a bit more shoulder room with 2 in it, but I don't know if that style is available anymore.
I also used seats often for shorter trips such as to creche. It was less of hassle in traffic, and I usually continued on to work so I didn't want to have to deal with storing the trailer at work.
You may or may not need new tyres for your bike. I'd just use what you have until they wear out. On the flats and downhills, there's not much difference rolling along, but you'll definitely notice the weight uphills, especially with 2 kids. At least you'll get stronger as the kids grow heavier.
As for use after the kids outgrow it, we didn't bother, but then we have panniers to carry gear (and the family station wagon).
It's a lot of fun. I have many fond memories of trips with the girls, well worth the money spent to enjoy family time together.
We have 2 tandems now, so the family bike time continues...