Something is wrong with your opamp, obviously.

I tested my third and last LM7171 I have. Turned the power on and it worked for 4 seconds, then I could see on the scope the signal rising on the screen, (same pk-pk) it all look good just rising, when it got maybe 10 volt high the input and output signal went crazy I could see whole lot of noise, barley seeing my original signal.
Let it cool, Tried all my 3 chips and same garbage noise again. Works fine with a LM741.

 

I tested my third and last LM7171 I have. Turned the power on and it worked for 4 seconds, then I could see on the scope the signal rising on the screen, (same pk-pk) it all look good just rising, when it got maybe 10 volt high the input and output signal went crazy I could see whole lot of noise, barley seeing my original signal.
Let it cool, Tried all my 3 chips and same garbage noise again. Works fine with a LM741.

I have ordered some lm7171 dip format. See what happens with that.

 

I don't get when you say to feed a signal at W. W is the output. Can you explain?

Yes, W is the output of the modulator, it is also the input to your Opamp stage (thru the series resistor).

So you can disconnect the W from the modulator chip and connect that point to a signal generator.
(At the free end of the input series resistor). Effectively you are testing your Opamp stage in isolation then.

Since LM7171 is very high speed, I suspect it may be oscillating at some high frequency (perhaps not seen on scope?).

Also, if you have capacitive loading at the -ve input of the Opamp (pin 2), the risk of such a thing will crop up. Breadboarding can lead to all sorts of parasitics, that is why the possibility exists.

To check out, try placing a 4.7 pF capacitor across the feedback resistor going from Opamp output to the inverting input. This will change the frequency response but if it improves stability then you can give it a shot or tune it further.

P.S. I haven't been visiting here regularly, hence the delay in responding