If it's a food grade drum, that's probably not paint on the inside, but a food safe liner. It doesn't strip or burn off easily. Quickest way without a lot of elbow grease on your part would be sandblasting inside and out.
I for my drum had nothing on the inside, but on the outside I used "duxola" a paint remover gel you brush on and just let it work for 10-25 min and airless spray with water. I had to pay 37$ it "melts" the paint so you don't have to work - and best of all, non toxic.
I went with the sandblasting too and was real happy with it. Make sure you run a nice long smoke to season it before you cook on it. I ran 2 smokes thru it for good measure.
I have the nastier red liner, so will sandblast once I'm ready to do the full build (don't want to leave naked steel exposed to rust too long).
People do burnouts with scrap wood or a weed burner... but that always looks like an awful mess of char and ash and remnant liner / paint that didn't come out, leading to wire brush, orbital sander, power washing, old fashioned elbow grease... ugh.
I had a conversation on BBQ-Brethren with some guys that used Kleen-Strip paint / furniture stripper. Reportedly it worked great on outside paint and interior tan liners. I tried it on the red and no luck. Unlike Alf's product, it is toxic, so lots of washing required.