Has anyone had any success playing HOMM 3/4 on daily basis? Meaning that you get to play one move and your opponent then has X days to make his. So that you do not have to play at the same time, such game could last X months. Similar to playing chess on chess.com with 24 hours to make a move.
If you're asking if you can play a multiplayer game this way, then what you're describing is a multiplayer mode commonly called Play-by-E-Mail (PBEM). Unfortunately, no, the Heroes of Might and Magic games don't really support this mode of play.
While the HOMM games support multiplayer save games you can't a implement a PBEM game by exchanging save files without severe limitations. Battles between two players requires that they both be present simultaneously. Battles between AI players and human players require that the human player be present. When the AI takes its turn(s) this may mean that more than one human player needs to be present at the same time. You can work around this by auto-resolving all battles but then you've lost most of gameplay.
You could play a normal multiplayer game over months with long breaks between turns if you want, but if you want to play the full game both you and your opponent will need to be playing the game at the same times.
I should note that playing HOMM 3 multiplayer over the Internet is difficult these days. If you're playing the original version (including the GOG.com version) then you'll need to use special VLAN software that makes your opponent appear to be playing on the same LAN. The HD version of HOMM 3 on Steam has a more modern multiplayer implementation, but it not clear if actually works reliably.