Not going to work, at least not quite like envisioned.
The key to Hot air Ballooning, or other kinds of ballooning for that matter, is gas density. The gas in the balloon has to be significantly less dense than the surrounding gas. For something like a balloon suspended bridge, the density of the whole balloon bridge system has to be less than the surrounding atmosphere. That means total volume and mass has to be accounted for. Then it has to be less than the surrounding atmosphere. Here is where your first problem comes up. The air around lava is going to be hot, and not very dense. That means regardless of how you get hot air in the balloon envelope, You will still need a very large envelope.
Next, you don't want the envelopes, or even the bridge, too close to the surface of the lava field. Not only is the air being heated, it's also probably giving off all sorts of nasty fumes and such. These are certain to be un breathable and also likely to be caustic. Wood and natural fibers are going to rot quickly, requiring near constant maintenance. Using steel would slow the process, but it would be way too heavy to be practical.
Finally, you need to consider that the closer to the lava field, the hotter the air will be. It will cool quickly as it mixes with outside air, but if you are too close to the surface, anyone standing on the bridge runs a chance of being baked alive. How long could you stick your head in a 400 degree oven and breath that air? It would dry lungs out in a flash. Your bridge has to be high enough so that you don't cook your travelers.
So here is what might do. A bridge in need of enormous amounts of maintenance. You could protect the envelopes somewhat by affixing them to very long lines above the bridge. That would add to bouyancy as well. Downside to that is you would then add complications of different winds the higher up you go. The travelers would have to ascend to the bridge surface well back from the lava field. They would then have to descend a good ways from the field on the other side.
So it could work, but it becomes something so strange that simply going up and over in an airship would be simpler, safer, and much more certain. Kind of like the apocryphal story of the US spending a lot of time and development going into a zero gee pen and the Russians just using a pencil.