A chemical injection valve can include a valve stem, flow between sections of a flow passage being prevented and permitted in respective closed and open positions of the valve stem, a resilient primary seal that prevents flow through an annular gap surrounding the valve stem in the closed position, and a bypass passage in communication with the annular gap and a flow passage section, and in communication with another flow passage section in the open position. A chemical injection system can include a chemical treatment pumped through a chemical injection valve and into an interior of a tubular string, and the chemical injection valve including a valve stem and a bypass passage extending between a flow passage section and an annulus surrounding the valve stem, the annulus being positioned longitudinally between a resilient primary seal and a metal secondary seal.