First of all, the "frameless rendering" technique is in the context of raytracing, not rasterization. It's not obvious how it could be made to work effectively with rasterization, given that the basic idea of it is to update an image by a combination of temporal reprojection plus firing rays specifically at areas where the algorithm thinks the image is undersampled.
So this technique is, prima facie, not compatible with rasterization-based graphics applications. But if you're already doing raytracing for other reasons, this technique would be interesting to look at; it certainly appears to improve quality relative to an image raytraced from scratch each frame, with the same number of rays per second.
Raytracing on a GPU is certainly possible; you don't need an FPGA for that. I've only skimmed the second paper, but my reading of it is that the main reason for the FPGA is to get a close coupling between the display scanout and the rendering activity: namely they "race the beam" and evaluate pixels just before they're about to be scanned out, thus obtaining low latency.
The GPU equivalent of this is probably to split the image in thin horizontal strips, and kick off a compute dispatch to render each strip just before it starts to be scanned out. Today, this is difficult to accomplish as it requires either millisecond-precise scheduling that desktop OSes are not currently set up for, or it requires the GPU to be able to dispatch based on an interrupt from the scanout unit—a hardware feature that doesn't currently exist AFAIK (or if it does, it isn't exposed in any graphics APIs). Or you might be able to make it work with a long-running asynchronous compute dispatch, if you can find a way to stay in sync with scanout.
So, there are obstacles, but they aren't insurmountable, and I think if there was sufficient interest in racing-the-beam-style rendering, then OS and GPU vendors could come up with a way to do it in the future. So I don't think an FPGA is required for this kind of technique to work. On the other hand, the fact that it's based on raytracing is a much bigger obstacle to using it in "real-world" apps and games.