It depends on the projector, which model of Fury, and what you're trying to do. The Fury is a format converter only, HDMI to RGBHV, so resolution in = resolution out.
Some CRTs like the BArco 1209s and the NEC XG will show image wraparound at 1080p, so you'll need a scaler to adjust the image to get rid of that. If you want a 1080i signal upconverted to 1080p, you'll also need a scaler for that. But, in most cases you can go right into the Fury and out to the projector.
Also note that the Moome external box has better signal processing than the Fury.