In the sport of Jiu-Jitsu, close contact with many partners is common. Many times these partners are sweating profusely (Yuk), which in turn drips on the other person sometimes into the eyes, nose, and yes mouth. I am just wondering if viruses can enter in this way.
Sweat per se, at the source, is not likely to be contaminated with bacteria or viruses. However, that's a moot point, because as soon as it's produced it's on contact with the sweaty person's skin, which is invariably coated with bacteria and potentially with viruses. Sweat can therefore act as a medium for transferring infectious agents between two hosts.
Examples of this are described in Contact transmission of vaccinia virus from smallpox vaccinees in the United States, 2003–2011:
Among adult male cases, the most frequently reported activities associated with vaccinia transfers were wrestling, grappling or sparring (n = 18) [12,18–20]; household interactions (e.g., sharing towels, clothing) (n = 6); unspecified “contact” (n = 4); and football or basketball (n = 3).
(Vaccinia virus is the vaccine for smallpox. As such it's far less dangerous than smallpox itself, but as an ancient vaccine it's also much more dangerous than the vast majority of modern vaccines, and exposure to large concentrations of the virus in an uncontrolled setting is potentially risky.)
In short, yes. Virus particles, or bacteria on one persons skin can end up being in the sweat that is transferred from one person to another. They can also be transferred by direct physical contact though, so the sweat isn't the only source of transfer.
I will caveat that a healthy person shouldn't need to worry about the amount of harmful bacteria in sweat as the numbers will be so low that their immune system can take care of it fine.