How can I build a safe trash can/incinerator?

Make a toilet!

This way, nothing gets incinerated until a player activates it. I recommend using a button instead of a lever. That way, it resets itself.

You can always just drop items in a 1-block-high pit and ignore them, where they will despawn after 5 minutes and you won't pick them up unless you go into the pit. But if you really want to set things on fire, this will do:

That's a nether brick fence capping a lava pit. Items hit the fence and fall straight down into the lava, but the gap is not wide enough for any mob but chickens and silverfish to fall in. (The lava could be raised up one block if you want; I was designing this as I went and left it that way.)

This video tutorial I made (with co-conspirator Kaliard) demonstrates how to make a safe, easy-to-make, and stylish "trash can" incinerator using a chest, a hopper, and a dropper (those last two new in Minecraft 1.5) in a vertical stack, with a redstone clock to activate the dropper. The hopper pulls any items you put into the chest and puts them into the dropper, which ejects the items into a pool of lava in front of it when you throw the switch.

The benefits of this design are safety for you, your pets, and especially your items. There's no worry about accidentally hitting the "throw" key while holding your enchanted pick, as with open-pit designs. Because the redstone clock can be turned on and off with a switch, you can load up the trash can and be sure that, when you flip the switch, only the items that you intend to incinerate will be destroyed.