A non-resonant Beverage antenna shouldn't need a tunable matching network for either transmitting or receiving (but as Phil nicely stated, you certainly can use one). All you really need is a proper broadband matching transformer, 9:1 for a 50 ohm feedline and 6.25:1 for 75 ohm. It is better to connect the Beverage directly to it and then run coax between it and your station.
Phil is correct about the losses (my Beverages have about 10 dB of loss) and that it mostly futile to transmit on them. However, myself and other happy Beverage antenna users have successfully transmitted on them. Among other things, it's a matter of how the gain and takeoff angle of a Beverage compares with another antenna over a given DX path and band conditions.
For example, one afternoon last year I compared my low N-S 75m dipole with my 580' NE Beverage. I was in QSO with a couple of hams in New York State from my QTH in SW Missouri. I had to transmit on my Beverage because they could barely hear me on my dipole. On another occasion, I had a significant signal into Europe transmitting on this same Beverage compared to another antenna.
Usually, 40 or 30 meters is about the highest band that they are useful on, but there have been exceptions. W8JI told me that on occasion, his 580' Beverages heard a 20m DX signal that his high Yagi could not.