There are no restrictions in the US (and I doubt in other countries). Plenty
of doctors have chronic hepatitis, herpes, and any
number of infections. They are not routinely tested, and if tested and found positive, continue their training or practice without interference. As you will learn once you begin medical training, with easy, common-sense precautions, such doctors do not transmit these infections to their patients.
You can easily find oral herpes symptoms on line.