This scenario sounds like either the Ignitor (High Voltage Pulses - like an Automotive Ignition System), the Start Switch on the Blower Motor (on split phase Motors), or both, are causing the interference.
These sources will cause a series of pulse-like "streaks" to appear across the CRT along with an audible noise (normally) in the audio portion of the TV Receiver.
The interference could be entering the set from the Antenna line, the AC power circuit, or both.
As stated before, passing the noise to ground via filters will work. Find out where it is entering from, then filter that circuit at the set.
Also, be sure the metallic frame of the furnace is driven to ground (properly grounded). This will assist to shield some of the Capacitive bursts from the Ignitor or Motor.
P.S. If for some reason the Blower Motor is a Brush Type Motor (like a DC Motor), the brushes will create a continuous EMI / RFI situation on the CRT, which worsens as brush-commutator contact also worsens. The results of sparkee brushes will show up on CRTs.
To catch this interference, place a low value capacitor across the Motor's input leads - right at the Motor.
Feel free to ask further questions about this situation.
Scott " 35 " Thompson
Just Say NO To Green Eggs And Ham!