Hydrocarbon wells including gas lift valves and methods of providing gas lift in a hydrocarbon well. The hydrocarbon wells include a wellbore extending within a subsurface region and a downhole tubular extending within the wellbore. The downhole tubular defines a tubular conduit, and the wellbore and the downhole tubular define an annular space therebetween. The hydrocarbon wells also include a lift gas supply system configured to provide a lift gas stream to the annular space and a closure material supply system configured to provide a closure material stream to the annular space. The hydrocarbon wells further includes a gas lift valve operatively attached to the downhole tubular. The gas lift valve includes a lift gas injection conduit and an actuation mechanism. The actuation mechanism selectively transitions to a closed state responsive to contact with the closure material. The methods include methods of operating the hydrocarbon wells.