If you have imported your icons and set their paths correctly, they should always display properly in all circumstances. It sounds like your map's .mpq file may be unstable, or, perhaps the tavern has some bugs?
Some troubleshooting possibilities include:
- Try setting default icons (some of those included in Warcraft III's .mpq files) for those heroes.
- Rebuild the hero tavern from scratch.
- Check the map for any other imported content and/or triggers that might be causing complications.