In Final Fantasy VI Pixel Remaster, there is no locked house in Thamasa that you need to enter. However, Thamasa is a significant location in the game, and here’s how you navigate it:

### Investigating Thamasa

1. **Item Collection**:
– **Eyedrops**: In the middle barrel next to the Item Shop.
– **Gold Needle**: In the southern barrel alongside the Relic Shop.
– **Phoenix Down**: In the northern barrel near the Inn.
– **Echo Screen**: In the northern barrel of the northwestern house (the Mayor’s house).
– **Green Cherry**: In the northern barrel of the northeastern house.

2. **Relic and Item Shops**:
– The Relic Shop sells many of the best Relics available to this point, so buy whatever you may be lacking.
– The Item Shop sells various potions and other items.

3. **Meeting Strago and Relm**:
– Enter the northeastern house and speak to the man there (Strago) and his granddaughter (Relm). You’ll be prompted to name them.

4. **Preparing for the Fire**:
– Strip Shadow of any items you might want (especially his Esper and Relics).
– Go stay at the Inn for a bargain price of 1 Gil.

5. **The Fire Breaks Out**:
– Wake up and go to the huge mansion to the east for a scene.
– Be ready for a battle with Balloon monsters as you navigate through the burning house.

6. **Navigating the Burning House**:
– Use physical attacks to reduce MP drain against groups of three Balloon monsters, and magic against groups of six.
– Take the right door in each room to avoid unnecessary battles and find treasure chests containing items like the Flame Rod and Ice Rod.

7. **Finding the Memento Ring**:
– After navigating through the burning house, go upstairs in Strago’s house and search the wall next to the table to find the Memento Ring.

8. **Setting Out from Thamasa**:
– You’ll be upstairs in Strago’s house. The Memento Ring is available, but only Shadow can use it currently. Hold onto it as it’s incredibly useful.

### Conclusion

There is no locked house in Thamasa that you need to enter in Final Fantasy VI Pixel Remaster. The focus is on collecting items, meeting key characters, and navigating through the burning house to progress the story.