If the difference between futures and spot prices rises will the yield for the current bond increase as well?
The difference between futures and spot is related by the carry. Assuming a positive carry environment where the yield curve is upward sloping, then the spot price is less than the futures price.
If prices rise, the yields will change.
The cause-effect relation is the other way round.
The driver is the change in yields, and the effect is the change in the difference between futures and spot.
The fixed income markets (specifically the government bond markets) are among the deepest (i.e. highest volumes and notional balances) and most liquid.
I do not believe anyone tracks the difference between futures and spot (for any underlying security - stocks, FX or commodities) to take a cue that risk-free yields (govie bond yields) to react to a change in future-spot difference.
It is quite possible, for a variety of reasons that the future-spot difference changes while yields show no related changes - because the futures/spot difference can change to due to limitations in the liquidity of the underlying so that any arbitrage opportunity cannot be effected due to large transaction costs.
Lastly, when you say yields you need to be more specific as interest rates come is quite a few flavours.
The interest rate most relevant to spot-futures arbitrage argument is the repo rate. This is the rate which is likely to be most responsive to the change in the difference between futures and spot.
And one more really last thing, the arbitrage argument is really for the spot-forward-repo rate trinity. For futures, a convexity adjustment (on top of the forward price) is also relevant. So a change in underlying volatilities could theoretically increase the futures price (while spot and repo rates remain unchanged). Although, to my best understanding is will normally be quite small to bother.