What is the response current of a photodiode to an LED emitting light at different intensities?
at a given time, the same LED is either on or off, and to virtually get different intensities, the duty-cycle of being turned on and off has to change. So, the response is like ones and zeros.

No, that's incorrect.
The amount of photons emitted from an LED is directly proportional to the current through it.
It's a common method of varying the perceived brightness by using methods like PWM, yes, because it's energy efficient and easy.

Will the photodiode's response be ones and zeros too?

No.  The more photons hit it, the higher the current is going to be, so the photodiode itself has a continuous response.
Now, when being illuminated by an off/on pulsed source, it depends on whether that source is pulsed faster or slower than what the bandwidth of the photodiode allows.
Photodiodes are available in bandwidths from Hertzes to hundreds of Gigahertzes, so there's no general answer here - but if it's something with through-hole leads, don't expect more than a couple hundred kilohertz of bandwidth, if at all. Again, you'll really need to read the datasheet of the photodiode, no way around that.
If the bandwidth of the light signal is higher than that of the photodiode, that will have a low-pass effect and lead to a &quot;smooth&quot; signal with arbitrary amplitude.