Do the birds just attack your seedlings, or have they eaten the vine tips and/or pods? If you've had problems with birds attacking the plants when mature, there will probably not be a safe time to remove the cover.

However, peas like cool temperatures, and one good sunny day could bake them, or at least cause blossom drop. I'd recommend replacing the clear (and unbreathable) plastic with floating row cover. Use the lightest fabric weight possible. This will keep out the birds, but should cause only a minimal temperature increase under direct sunlight. I've done this on my chard & peas to keep birds from damaging them, with great success. As an added bonus, the cover keeps out most bugs too.

Birds don't attack mature vines, but they will eat the peas in the ground, and I assume they would pull up small seedlings. The peas are now just out of the soil and have not completely unfurled their true leaves. I'd like to use a row cover too, but I am kind cheap, so if the birds and heavy rain are no longer a problem I'll probably just remove all covers.

I have found this nylon French Tulle, to protect my seedling from squirels, rats, rabits and birds. Those creatures won't want to bother with the tulle. I also use finer mesh of the same to protect my eggplants from those nasty flies/tics that perforate eggplant leaves. The stuff is inexpensive, light weight, wind/rain proof. You find it at any fabric store. I buy them from Hobby Lobby.