As long as you use software encoding, it should make no difference if Intel, AMD or whatever. There are of course slight differences in performance because of the different chip designs. You need a powerful enough CPU if you plan on streaming full HD 60fps of course, so an old AMD may not be sufficient whereas a new Intel would be fine.
If we're talking hardware encoding, then it's a different story. Modern Intel CPUs come with QuickSync (QSV) as hardware encoder for H.264, some of them also support HEVC / H.265 and partially VP8/VP9. In OBS, there's only support for QSV H.264 at the moment however.
Hardware encoding means it is very efficient and does not really steal power from your CPU or GPU if you encode while gaming for instance. The disadvantage is that it is an actual chip, which can not change like a software encoder can. It is not capable of producing the best visual quality per bitrate (at least not the encoders built into CPUs/GPUs), whereas a software encoders can be tuned to give better quality at the expense of more CPU load.
That said, software encoding is usually the better choice for live streaming if you have a fast enough CPU which can handle a game and the encoding simultaneously - it gives a better quality for typical bitrates compared to hardware encoded feeds.
AMD also offers support for hardware encoding. It is called Video Coding Engine (VCE). OBS Studio does not ship with support for this, but there's a plugin to enable it:
https://github.com/Xaymar/OBS-AMD-Advanced-Media-Framework
It is under active development, but don't know what works and what does not at the very moment, as I don't have AMD hardware to test it myself. I'm actually not sure if both, CPUs and GPUs by AMD support VCE, but it's possible.
If you have an Nvidia GPU, then you likely have support for their hardware encoding too, called NVENC. OBS Studio supports it out of the box.
Between all these hardware encoders there are certainly differences in quality etc., but there's no general rule which one is the best, as the chip design may differ between every CPU/GPU model. For instance, Intel's Skylake series is said to produce higher quality videos at the same bitrate than older series like Haswell. At high bitrates for offline recording, there's probably not a huge difference in the end. QSV using LA-ICQ 10 (lookahead 100) gives extremely good quality, but also results in pretty large files (50GB/hour for 1080p@60, Battlefield 1 Beta). The tricky part is actually to find a middleground between quality and size.