It would help to know the tray material and finish (a picture of the edge detail would be great), and it isn't clear whether you are referring to just the outer edge of the mosaic area or every glass tile, but here are a few ideas:
If the problem is the outer perimeter of the mosaic area, remove the sharp edges with a Dremel tool and diamond disk. Clamp the tray and tiles in a towel to dampen vibrations. Use a little water or oil on the spot you're grinding to reduce heat and vibration. You won't need to remove much to dull the edge.
If the problem is the outer perimeter of the mosaic area, cover the edge with clear silicone caulk. Clean the surface with alcohol. Use masking tape to create a clean, straight border close to the sharp area on both sides of the mosaic edge. Either use your finger to spread the caulk, or dilute the caulk with paint thinner and paint over the sharp edge using a brush. Remove the masking tape while the caulk is still uncured. I would be careful using the tray because silicone can often be peeled off surfaces it's adhered to (but if that happens, you can peel it off and replace it).
If the problem is the outer perimeter of the mosaic area, an alternative to a clear covering would be to cover the sharp edge with a "leaded glass" border. Wrap the sharp edge with a strip of copper foil, then solder the foil.
If the problem is on all of the glass tiles, you could add a layer of clear silicone over the whole surface. However, sharp edges over the entire surface would indicate that the grout isn't high enough. You can get color-matched grout at big hardware chains or stores that sell tile. Add more grout to bring the level up to the top surface so no sharp edges are exposed. That might also be an option if the problem is only on the outer perimeter; depending on the design of the tray, you may be able to add a border of grout around the outside edge of the mosaic to cover the sharp edges.