An earth like planet in terms of lots of water and an atmosphere. The liquid water on the sunny side would evaporate due to the extreme heat and the snow and ice on the dark side would never melt due to lack of ever getting solar energy. So would the build up of solid h2o on one side of the planet prevent the planet from maintaining a geosynchronous orbit?
If it could maintain a synchronous orbit, what would a planet's weather patterns look like?
Yes.
Water evaporating off the dayside and permanently condensing on the night side, resulting in a desiccated dayside, is one potential outcome, but only for a certain range of atmospheric density and composition and water content.
It actually takes surprisingly little atmosphere to distribute enough heat around a tidally-locked planet to ensure that liquid water can still exist on the night side. But even if you don't have that, a sufficient quantity of water will ensure that glaciers completely covering the night side will nevertheless creep back around into the light where their edges melt and re-introduce liquid water to the dayside.
As for what the weather will look like, the details are highly sensitive to things like exactly how much atmosphere there is, how fast the planet is rotating (equivalently, what its year length is), how much surface water there is, etc. In broad strokes: it is very easy to end up with either a single permanent cyclone over the sub-stellar point, or a pair of permanent counter-rotating cyclones on either side of the equator; it is also fairly typical to end up with high-altitude super-rotation of the atmosphere (similar to what we see on Venus), while surface-level prevailing winds tend to run from day to night along the equator, and from night to day over the poles.
Slightly more details can be found in, e.g., Simulations of the Atmospheres of Synchronously Rotating Terrestrial Planets, or any of a surprisingly large number of related papers.