No. Most modern DVRs have their own tuners (either pay TV or OTA) and as such, don't work with external tuners. Even if they did, guide data for FTA isn't available so the DVR can't control the FTA receiver.
The Linkbox 9000i Local is the only box I know of that will do both OTA and FTA (two tuners). Connect one input to an antenna, one input to a dish.
Like Harshness said, you will not get any channels the cable companies get. FTA is completely different. Lots of variety, different stuff (and some would say better stuff) than what you get with cable or pay-satellite.