Will overlapping two different beams of coherent light with different wavelength cause interference?

Yes, they will of-course overlap but it won't be the same pattern for when waves with same wavelength overlap. The resulting wave can be found from graphing simple addition of sine waves indicating the superposition at each point.

To explore this visually you can try the graphing calculator Desmos. Try changing the slider values of $a$ and $b$ in the resultant wave,$y=sin(ax)+sin(bx)$ to change the wavelength of each wave and see what happens to the wave after interference(producing resultant wave).

Note: They will only interfere if they meet/cross paths in the optical fiber. Also you may try change the phase difference by a third slider e.g. $k$ in $sin(ax-k)$.

Interference is a concept that only has true meaning when comparing two signals of the same wavelength/frequency.

For waves with different wavelength, it is true that the snapshots of the electric field (or magnetic field for that matter) will change because of the two signals, but there is nothing coherent about such addition.

By coherent, it means that the interference persists spatially and/or temporally.

Consider two plane waves with frequency $omega_1,omega_2$ and phases $phi_1,phi_2$. Examine their total intensity:

$$I_{text{coherent}}=|e^{i(omega_1t+phi_1)}+e^{i(omega_12+phi_2)}|^2=2(1+,text{cos}(Deltaomega t+Deltaphi))$$

$$I_{text{incoherent}} = |e^{i(omega_1t+phi_1)}|^2+|e^{i(omega_12+phi_2)}|^2=2$$

Notice that the intensity for an incoherent intensity, the sum is simply "$2$", but if interference is taken into account it deviates from that value. However, that deviation will rapidly vary in time if there is a frequency difference between the two beams $Deltaomeganeq0$, and will be washed away if you average in time. This is why you don't get "interference" between two wavelengths of different values, the interference averages to zero.

However, if you were to look at a single moment in time, yes the intensity will fluctuate on the scale of $1/Delta omega$, so this does matter when you do time-resolved measurements.

Yes, light from the two lasers will interfere, but the light will interfere temporally, not spatially. The result, when the light is incident on a detector, is a beat frequency: a signal whose frequency is equal to the difference between the frequencies of the two lasers.
The phase of the signal will wander relatively slowly, at a rate that depends on the phase stability of the two lasers.

The beat frequency between two lasers having slightly different frequencies is temporal interference, and is actually exactly the same phenomenon as spatial interference. A fringe pattern that is stationary in one reference frame is moving in another. The frequency of both beams is the same in the first frame, but different (because of Doppler shift) in the second frame, the rate at which the fringe peaks go by in the second frame is the beat frequency between the beams: one Doppler shifted up, and the other Doppler shifted down.

If the lasers are incoherent then there will be no interference between the two signals. If they are fully coherent then they will fully interfere. In the latter case it cannot be determined which laser emitted a particular photon. The situation is very similar to two slit interference. Each photon is emitted by both lasers as far as quantum physics can tell!

Consider 2 lasers (blue and green) shining on a wall (rough surface) on the same spot. You basically will see 2 interference patterns (called laser speckle) overlapping but they do not interfere with each other. You can shine one laser at time and add the 2 pictures separately to get the same pattern as you would with shining both lasers.

Take a double slit and the same is true, you will see 2 patterns that are just an over lap of each other.