I have encountered a few bugs with Burn Up and the different interactions with transform compared side by side from retail to showdown. Special thanks to Nuren for helping me test this.
Retail Pokemon Battle: HEKG-WWWW-WWWG-5AHB
Turn 1: Wyrx: smeargle uses splash Nuren: arcanine uses burn up
Turn 2: Wyrx: smeargle uses transform Nuren: arcanine stalls with burn up
Turn 3: Wyrx: smeargle-arcanine stalls with burn up Nuren: switches out to smeargle(edited)
Turn 4: Wyrx: smeargle-Arcanine uses burn up (fails) Nuren: Smeargle uses shadow ball (no effect) >>The transformed arcanie is pure normal type
Turn 5: Wyrx: smeargle-arcanine switches to smeargle Nuren: smeargle switches out to arcanine
Turn 6: Wyrx: smeargle uses forest's curse (before burn up) Nuren: arcanine uses burn up
Turn 7: Wyrx: smeargle uses transform Nuren: arcanine stalls with burn up
Turn 8: Wyrx: smeargle-arcanine uses burn up (nothing happens) Nuren: arcanine switches out to ho-oh
Turn 9: Nuren: ho-oh uses burn up and smeargle-arcanine faints (SE hit) Wyrx: n/a, sends out new smeargle >>smeargle-arcanine became at least part grass type(edited)
Turn 10: Wyrx: Smeargle uses transform into ho-oh Nuren: Ho-oh uses burn up (nothing happens)
Turn 11: Wyrx: Smeargle-ho-oh uses shadow ball (neutral damage) Nuren: ho-oh uses shadow ball (neutral damage) >>neither is normal type
Turn 12: Wyrx: smeargle-ho-oh uses burn up (nothing happens) Nuren: Ho-Oh stalls with roost
Turn 13: Wyrx: switch to smeargle Nuren: Ho-Oh stalls with roost.
Turn 14: Wyrx: Smeargle uses transform into ho-oh Nuren: ho-oh stalls with roost
Turn 15: Wyrx: smeargle-ho-oh uses burn up (nothing happens) Nuren: Ho-oh uses shadow ball (neutral damage) >>transformed burned-out ho-oh is not fire or normal type
Turn 16: Wyrx: smeargle-ho-oh stalls with roost Nuren: ho-oh switches to arcanine
Turn 17: Wyrx: smeargle-ho-oh uses giga drain (NVE) Nuren: Arcanine uses burn up (neutral)
Turn 18: Wyrx: switches out to smeargle Nuren: Arcanine uses burn up (nothing happens)
Turn 19: Wyrx: smeargle uses forest's curse Nuren: Arcanine uses burn up (nothing happens)
Turn 20: Nuren: arcanine switches to ho-oh Wyrx: smeargle uses forest's curse
Turn 21: Wyrx: Smeargle uses splash. Nuren: Ho-oh uses burn up.
Turn 22: Wyrx: smeargle uses transform into ho-oh Nuren: Ho-oh stalls with roost
Turn 23: Nuren: Ho-oh switches for smeargle Wyrx: smeargle-Ho-oh stalls with roost(edited)
Turn 24: Nuren: smeargle uses spore (nothing happens) Wyrx: ho-oh uses roost to stall >>transformed burned out ho-oh hit with forest's curse remains grass type
Turn 25: END
Differences in showdown. Note: Turn 9 the turn order gets shifted around because Smeargle didn't die due to incorrect burn up mechanics.
https://replay.pokemonshowdown.com/gen7customgame-697780715
Turn 1: Same as retail
Turn 2: Smeargle became a ??? type (Showdown) and not a Normal type (Retail)
Turn 3: Same as retail.
Turn 4: Smeargle hits Arcanine with shadowball(showdown) doesn't affect Arcanine(retail)
Turn 5: Same as retail both parties switch.
Turn 6: Smeargle used Forrest curse. Arcanine loses grass type + fire type (showdown) Arcanine loses Fire type and keeps Grass type (retail)
Turn 7: Smeargle transformed into Arcanine as a ??? only type (showdown) Smeargle becomes a Normal / Grass type (retail. tested in another battle)
Turn 8: Same as retail.
Turn 9: Ho-Oh hit smeargle-arcanine for neutral damage (showdown). Ho-Oh hit smeargle-arcanine for super effective damage + KO due to forest's curse (retail)
Turn 10: N/A had to switch due to faint in retail vs non faint in showdown.
Turn 11: Same as retail (Turn 10)
Turn 12: N/A incorrect move usage (used burnout to test if fire type)
Turn 13: Same as retail (Turn 11)
Turn 14: Same as retail (Turn 13) refer to the burnout test of Turn 12.
Turn 15: Same as retail (Turn 14)
Turn 16: Same as retail (Turn 15)
Turn 17: Same as retail (Turn 16)
Turn 18: Same as retail (Turn 17)
Turn 19: Same as retail (Turn 18)
Turn 20: Same as retail (Turn 19)
Turn 21: Same as retail (Turn 20)
Turn 22: Ho-Oh used burn up and became a ??? flying losing both its fire and grass typing from forest curse (showdown). Ho-Oh used burn up and became a grass-flying (retail) (Turn 21)
Turn 23: N/A Incorrect move usage
Turn 24: Smeargle transforms into a ??? flying type (showdown) Smeargle transforms into a Grass / Flying type (retail) (Turn 22)
Turn 25: Same as retail (turn 23)
Turn 26: Smeargle uses spore and puts Ho-Oh to sleep (showdown) Smeargle uses spore and nothing happens (retail turn 24)
Battle Ends.
Bolded the differences between the retail and showdown battle. TL;DR Burn up incorrectly removes the additional typing of forest's curse on both dual types and mono type pokemon. Having a pokemon transform into a mono-type pokemon that used burn up will result in a normal type not a ??? type.