Does the Great Pacific Garbage Patch increase sedimentation rates over the patch?
This is somewhat a partial answer, as there is still a considerable amount of research being performed on the effects of the Garbage Patch.  The focus of this nswer will concentrate on what the US EPA (3) identifies as being the pollutant that contributes 96% of the Garbage Patches in the north Pacific - 'microplastics', pellets less than 5 mm in size.

According to Rios et al. 2010 (1), the plastic in the North Pacific Gyre are found to function as "a solid-phase extraction media" causing adsorption and concentration of pollutants out of the water column, similar to what has been found for marine sediments.

However, microplastics are found to be what Rios et al. (1) describe as being "inverse sediments" in that they do the opposite of natural marine sediments and remain floating on or near the ocean surface. Zarfi et al. 2011 (2) observed that microplastics can be incorporated into the natural sediments, and potentially wash up on beaches (e.g. if caught in a coastal current as seen n the diagram below) etc - whether or not, being incorporated into these natural sedimentation processes provides a 'sink' for the microplastics is still unclear (2). 

References

(1) Rios et al. 2010, Quantitation of persistent organic pollutants adsorbed on plastic debris
from the Northern Pacific Gyre’s ‘‘eastern garbage patch’’, Journal of
Environmental
Monitoring

(2) Zarfi et al. 2011, Microplastics in oceans, Marine Pollution Bulletin

(3) US EPA, 2011, Marine Debris in the North Pacific, U.S. Environmental Protection Agency