Yes, of course most of the parks you can't practically close, but you can officially close them - close the road gates, shut down the services, etc. In some cases, that's not a giant obstacle to using the park (a lot of the parks here on the east coast can be walked into from not too far away), but someplace like Yosemite it would be a different story. And some of them they can't fully close anyway because there are communities within the park - can't deny residents of the park access to their own homes. I'm sure they'll work something out - probably have to have the gates manned to turn away non-residents. If you want to see some amazing parkland in that part of the country, try Capitol Reef - it is cris-crossed with public roads. The only thing would be the visitors centers would be shut down.
When some parks get closed in winter I'd either bring my mountain bike or snowshoes and just park at the gate and enter. In summer, why the hell not just take a bike from the gate and have the pace to yourself. If the govt does shut down, I might head up to Arches to see it for the first time ever without people!
I highly doubt a trespassing charge would hold, especially in a place like Utah. You would still have to get caught, and then, have something debatable in terms of legality to have anything hold up in court. "Sorry sir, you can't take pictures in this park today. Our government is shut down". --yeah right.
Alright, then I'll shut up.
It's better to ask forgiveness than permission, dude.
ESPECIALLY true with photography, I've found.
Under normal seasonal closures, the Tioga Road is gated just above Crane's Flat. It would be great to bicycle up the road, but alas, the fine makes it not worth the fun.
Now Redwood National Park would be near impossible to close. One dead-end road is gated, but there are no other gates. And fortunately there are the surrounding State Redwoods Parks with access (but who knows what the State budget mess will result in.)