How would plate tectonics differ if Earth had no water?
One way of looking at this question is to determine if Earth would possess an active-lid (multiple plate), stagnant-lid (single plate) or an intermediary state (e.g. episodic or sluggish) tectonic regime.  This is one of the reasons comparisons are often made with Venus, especially in regards to the main difference been that Venus has a stagnant-lid with possible episodic active tectonic regime (see question and answer to Can Venus be considered to be tectonically active?, compared to Earth's active-lid tectonic regime.  

However, there are other differences between Venus and Earth that also may account for the differences in tectonic regimes, asides from the relative absence of water on Venus. Some modelling by Bouffard, (2013) indicated that different plastic yield strengths of lithospheric rheology could account for the different tectonic regimes.

How water interacts oceanic lithosphere is a vital consideration for this kind of question, considering that most spreading ridges and all subduction zones are based on the oceanic lithosphere. 

This answer is based on how the absence of water may have prevented a key theorised tectonic mechanism from even starting.

A crucial role for water is modelled by Regenauer et al. (2001) for the initiation of subduction and as they assert the beginning of plate tectonics, based on the theory that long term build up of sediment at a passive continental-oceanic boundary (e.g North American east coast) may have caused the strain to build up to a point where the oceanic crust 'breaks' and starts subducting, as shown in the diagram below:



Image source

In the absence of water, this process of subduction initiation (should it be true) could not have started; as the authors believe could be the case for Venus where volcanism is evident, but the mechanism appears to be hotspot activity through the current stagnant-lid tectonic regime observed there - if this theory were true, then the absence of water would mean that Earth would potentially have had a stagnant-lid, punctured by hotspot activity.

References

Bouffard, 2013, Role of water in the tectonics of Earth and Venus, University of Lyon

Regenaeur et al. 2001, The Initiation of Subduction: Criticality by Addition of Water? Science