I suspect strongly that the valve is gummed up with sealant. From the photo it looks like you may be able to remove the "core" of the valve. That would allow you to either clean it, or replace it. The core is the silver part that is screwed into the brassy looking part.
Otherwise, if you can't blow the blockage into the tube with a pump, and the tube wont deflate through the valve if you hold it open, then it's time for a new tube.
For what it's worth, tubeless tires with sealant work much better than tubes with sealant. They do sometimes suffer this problem too, but in that case you can remove the valve completely and clear it by poking a spoke, nail, or stick through it.