Could we sequester CO$_2$ by dumping forests in deep oceans?

So, there's really two questions here.

1) if we were to take wood and store it in a way that prevents decomposition, would this help reduce atmospheric CO2? and

2) Would sinking it into the bottom of an ocean trench achieve this?

So, first things first, yes removing wood from the contemporary carbon cycle does reduce atmospheric CO2. We already do this (at least temporarily) by using timber for making things - depending on the type of product this has a half-life of anything from a couple of years (paper or biomass fuel) to about 50 (for sawlogs). You can achieve a similar effect by planting new forests - so carbon is locked up in living biomass.

We could come up with a more permanent solution, and this would have to be economic (and not require more CO2 emissions to accomplish than we saved in the first place). Current research is looking at sequestration of carbon as charcoal (as this is more biologically stable than wood which is a source of food for a lot of species). This can either be spread on the land as biochar, or I suppose buried terrestrially - I'm thinking of abandoned coal mines.

I'm not aware of deep sea storage as a widely considered option, and I suspect that this is mainly due to the techno-economic cost, and lack of research - the debate is still ongoing for many other more established carbon sequestration technologies. I don't believe there has been much investigation into the stability of wood in deep water (though it lasts for thousands of years in anoxic peat bog conditions - so I see no obvious reason why it wouldn't work).