I'm pretty sure this has been covered a bunch of times before so I'll make this quick.
Burn cards are only burned to address a very specific live poker issue - marked cards. There's no such thing as a marked card online so there's absolutely no point in replicating burn cards online.
To demonstrate how silly this is, online poker "dealers" don't use a cutting card either, but live poker dealers
do. Should we write and complain to the online poker sites about how their deals are wrong because they don't use a cutting card?
Folded hands should never
be put back into the bottom of the deck either*. There's a seperate muck pile for them.
Continuous shuffling makes absolutely no practical difference to the outcome of a hand. The flop, turn and river are all random in a live game. They're random in an online game with a set deck. And they're random in an online game with continuous shuffling.
* Or at least not in hold 'em or stud games. Some draw games allow folded hands to be shuffled back into the deck if you run out of cards during one of the drawing rounds.