Once Endless Riches is played, will you ever draw a card, or is burn the only way to fill the discard after it is played?

Yes, you can still draw cards, and yes, Burn is essentially the only way to fill your trash.

Endless Riches only skips your Draw Phase (the one-per-turn draw in your Beginning Phase). Other card effects that say "Draw X" still work normally — you just don't get your free draw each turn.

As for filling the trash: Endless Riches' replacement effect says "If a card would go to your trash from anywhere other than your Main Deck, banish it instead." So discarding from hand, killing units on board, and spells resolving from the Chain all get banished — none of those go to your trash. The only exception is cards moving from your Main Deck to the trash. Burn (which takes cards from the top of your Main Deck and puts them in trash) is the primary way to do that, and the ETB of Endless Riches itself includes a Burn 7 to get you started. Any other effect that moves cards from Main Deck → trash (not necessarily labeled Burn) would also work, since the origin zone is still your Main Deck.

One extra note from FAQ #11653: tokens are not cards, so the replacement doesn't apply to them — but tokens that are killed still vanish immediately after hitting the trash per token rules, not contributing to your trash either.