Can a runner trash his own resource?

Not without card that lets them do it. Like Aesop's Pawnshop

Also note that there is absolutely no limit on how many resources the runner can have, so it's not something you want to do anyway.

Although some resources are limited to one e.g. you may only have one console in play, so you could trash your "Grimoire" at Aesop's for the credits before installing "The Toolbox". Aesop's is currently the only way to trade up consoles, unless you are tagged and the Corp is very obliging…

Metropolis Games said:

We should try to be a little more consistent in the terminology we use when answering people's questions.

Grimoire and The Toolbox are not Resources…they are Hardware. While Hardware, just like Resources, can not be trashed by the Runner without the aid of other cards (Aesop's Pawnshop), Resources can be trashed by the Corp if the Runner is tagged. Hardware can not be trashed by the Corp this way, so in your example, even an obliging Corp player would not be able to help the Runner change consoles.

And just to be complete, Programs CAN be trashed whenever the Runner is installing new Programs, but only then.

Side Note, Lukas was quoted on a thread in the BGG forum commenting on the Console situation. While the game as it is now does not provide any way to change out consoles short of pawning it to Aesop, he said it would be in the spirit of the original game to allow players to "over-write" the old console with a new console (I'm paraphrasing a bit here), so we might see something in the FAQ about this.

In regards to Lukas talking about the console situation when I emailed him and asked him directly about it he said "The rules do not allow you to install another console if you already have one installed." I then pressed him to see if that was the intention or not and all he would say is "That is all I feel comfortable saying until the FAQ. Patience."

Toqtamish said:

The response someone on BGG got was, "This is actually a good question. The rules don't really allow for it, but it is certainly within the spirit of the original game. I'll have to think about it more but it will certainly be an FAQ entry."