I absolutely think they can so long as the intention of both parties is to just be friends. Add in anything different, and it gets sticky.
I have several guy friends and it's worked out just fine because even when we were all single, it was mutually understood that this was a platonic friendship. Without that mutual understanding, then no, I don't think that a "true" friendship can exist. Sure, feelings change and romantic love can grow out of friendship, but I think that at the point when the platonic feeling isn't mutual anymore, that friendship has turned into something different.
What's funny is that the intention really really seems to make the world of difference. With most of my guy friends, had they been interested in me, I may have at times been accused of leading them on - that's not because I was flirty in the slightest, but because I happily accepted offers to hang out, go hiking, whatever. Because a romantic intention wasn't there on the part of either of us, it worked out just fine and we're great friends.
I've found that guys are definitely different than girls when it comes to friendship. I don't mean to play into any stereotypes, but if I want to go shopping, watch a chick flick, or talk about people dramas, I would be inclined to go to my girl friends (or make the boyfriend suffer through it haha). If I want to hang out knowing that the conversation probably won't go deeper or get any more emotional than yesterday's weather, I'll choose to hang out with a group of guys. That doesn't mean that the friendships are worth any more/less, it just means that they're (in general), different. Sure, it's probably more of a culturally driven thing than an actual "person" thing, but it's there. Individually, I've noticed that everything is much more variable. But in a group situation, yes - my girl friends tend to be more into emotional discussion and my guy friends would rather be doing something without too much talk or discussing something technical.
Onto the opposite sex friends when in a relationship...
I have a boyfriend, and honestly, for me everything is situational. I wouldn't invite a group of guys to go hang out and not invite my boyfriend. I wouldn't schedule a time to hang out one on one just for kicks with those whom I see regularly. But then, I also wouldn't invite any group - be it girls or guys - to do something that my boyfriend would probably enjoy without at least throwing the invitation his way too.
I
would, however, catch up with an old friend who I might not see too often via a platonic lunch date or something. I
would go to an event to which I was invited but not my boyfriend, even if it includes girls and guys. I
would ask my good guy friends for help or offer one-on-one help with anything if it was needed. I
would be fine to hang out with my guy friends doing something that my boyfriend would enjoy if my boyfriend was busy and couldn't go. I
would be happy to hang out if I randomly happened upon one of my guy friends while around campus.
I just try to use judgment and avoid doing anything that I wouldn't want done to me in reverse.
Of course, I introduced my boyfriend to most of my guy friends and they hit it off to the point that they're some of his best guy friends now. Nowadays, more often than not, I'm the one getting left out when they want their video game "bro time."