TL;DR: No, you're absolutely not expected or required to do, bring, or wear anything at a screening of The Rocky Horror Picture Show
I don't see any way to answer this without reference to personal experience, so here goes.
Not bringing props:
Not a big deal. I've probably brought props about half of the times I've seen RHPS in the theater. Sometimes, the group in charge will have some extras for people who couldn't/didn't bring anything, and most regulars would be happy to share what they have if you ask. But if you don't bring anything and don't want to borrow/share anyone else's supply, it isn't a big deal. No one will mind at all. If I had to give a ballpark estimate, I'd say that in my experience, far less than half the audience actually brings props to most screenings.
These photos, taken in the theater during the song (There's a Light) Over at the Frankenstein Place reflect my experience pretty well. On screen, Janet (Susan Sarandon) is using a newspaper to keep the rain off her head, and it has become customary for the audience to follow suit. However, you can clearly see that most of the audience isn't putting newspapers on their heads.
Not dressing up:
Again, no biggie. The entire point of the RHPS audience participation phenomenon is to express yourself however you wish. That might mean dressing up as Frank N. Furter, or dressing up as Batman, or wearing your street clothes. It's up to you. No one will notice - let alone care - if you're there in jeans and a t-shirt. If you have an amazing Columbia costume or what have you, you might get some compliments, but if you're just dressed normally, you'll almost certainly be in the majority.
Again, these photos of audiences at RHPS screenings are a reasonable reflection of what I've seen in the theaters I've been to. A small number of fans in full Transylvanian regalia (usually sitting in the front rows), one or two folks in costumes with no apparent connection to RHPS, and a large majority wearing normal street clothes.
Not engaging in call-backs (i.e., shouting at the screen):
None of the things I've covered in this answer so far are big deals, but this one is the least big deal of all.
For starters, you're going to be in a dark movie theater, with a loud movie playing, and most of the audience will be screaming for most of the movie; their visual attention will be focused on the screen and the shadowcast below the screen. As such, it will be difficult for anyone to even notice that you're not shouting along with them. Just like every other time people go to a movie, they're watching the movie, not staring at the other people in the theater.
Furthermore, everyone was a RHPS virgin once. If you've never been to a screening, you don't know what to shout or when to shout it. You probably won't get a good grasp of the call-backs until your third or fourth visit. So there will always be at least a few people in the crowd who are just taking it all in, not participating in the madness.
This is a total non-issue. The shadowcast is determined long before the night of the screening, and at most theaters, there is a set group of people who serve as the shadowcast on a revolving basis. I've never seen the cast ask for volunteers at a screening.
Dancing the Time Warp:
This is the one aspect of audience participation that, if you choose to abstain from, you might catch some flack. At my first screening, I was a shy 12-year-old who was embarrassed to dance in front of strangers. The people around me realized this and encouraged me, and I quickly joined in - to much applause and cheering from the people near me.
Another time, I brought a date to RHPS. She had no idea what the movie was, or what would happen in the theater, and she refused to participate in any way. No one noticed that she was wearing regular clothes, not doing call-backs, not using props, etc, but when the Time Warp began, the people around us did react, and more than one person said stuff like "Get up and dance, dammit!", etc.
Even so, nothing will happen to you if you refuse to dance, but it may attract some attention from your immediate neighbors, and even a comment or two. Aside from these two instances, I've never noticed anyone not dancing the Time Warp; if you're shy or embarrassed to join in, no one will force you to do so, but you should try to remember that everyone is doing it, and you won't look any sillier than they do.
I think the reason people are more likely to comment when people don't dance, while they don't even notice when people don't do call-backs or use props, is because an RHPS Virgin has no way of knowing the call-backs or which props to use, but everyone knows how to do the Time Warp the first time around.
How do Virgins know how to do the Time Warp, you ask? Quite simply, because the first time the dance happens, the "Criminologist" narrator and the Transylvanians explain and demonstrate how to do it:
It's just a jump to the left
And then a step to the right! [repeated]
Put your hands on your hips...
And bring your knees in tight.
But it's the pelvic thrust... [backwards and forwards, repeating]
that really drives you insane! [in a circular motion]
Let's do the Time Warp again! [wave arms]