My daughter has this toy. Its a lantern with a dome over LEDs. The dome has holes in it in the pattern of constellations. So you turn on the lantern and it projects stars on the ceiling.
Why do the holes in the dome (slits in the screen) not cause an interference pattern?
The same question would apply to gobo's on stage lights.
The effect you are looking for is difficult to observe for a number of reasons.
There are four LEDs within the dome and hence this is an extended light source each of the LEDs may well be producing a visible interference pattern if they are small enough but with four light sources each of these interference patterns will overlap and so obscure each of the individual interference patterns.
With white light only a few orders can be seen because of the overlap of the interference patterns produced by each individual wavelength in the white light.
The intensity of your LEDs may be not great enough to observe the interference patterns.
If you have a laser pointer try and send the laser light through the dome with the light entering and leaving the dome at a small clear part of the dome.
You may well see an interference pattern?
Generally, you see interference and diffraction phenomena when the width of the slit $w$ is on the order of the wavelength of the light $\lambda$. Hence, if $w \gg \lambda$, you will not observe an interference pattern. Likewise, if $\lambda \gg w$, you will not observe an interference pattern. However, when $w \sim \lambda$, you may well observe interference phenomena.
In general the slits have to be very thin. This is part of the reason why interference and the wave nature of light weren't noticed during Newton's time. The slits have to be even smaller than what a razor can produce.
