Minecraft version: 1.20.1
Modloader: Forge
Forge version: 47.2.1
Environment: Singleplayer

First mod name: Cycle Paintings
First mod version: 3.2

Second mod name: Fast Paintings
Second mod version: 1.2.0

When right clicking with a painting in hand it doesn't do anything. I believe this is because fast paintings turns paintings into blocks to help with performance.

The cycling does work without issues, if the painting was placed on the same block as an entity (like a flower pot), and it keeps working even if said entity is removed after. Like so:

The cycling does work without issues, if the painting was placed on the same block as an entity (like a flower pot), and it keeps working even if said entity is removed after. Like so:

Looks like that's because the painting will remain an entity when placed next to a flower pot. (From the authors CF page)

Not an easy compatibility fix unfortunately. I'll take a look when I can, thanks for giving the mod a try and opening the issue!