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What would a red object look like shone onto a blue light?

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Red and blue are at opposite ends of the visible light spectrum, red being of low frequency, and blue being of high frequency. Sunlight contains frequencies across the whole frequency range, which appears white. An object that appears white reflects all frequencies.

A red object appears red because it reflects red (low frequency) light, and absorbs all other colours. Blue light contains only high frequencies. If you are in a darkened room, lit only by blue light, white objects will appear blue because they are reflecting the blue light, and no other frequency is available.

But a red object absorbs high-frequency (blue) light, and since this is the only light in the room, it does not reflect any light. Objects that do not reflect any light appear black. Some objects appear black even in sunlight, because they absorb all frequencies and do not reflect any.

So to answer the question, when you have a red object that is lit only by blue light, it will appear black.
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What colour would a blue object look like if a yellow light were shone on it?

It's called "A Blue Object" because when light of many colors falls on it, it reflects the blue light toward your eyes and absorbs all the other colors, including yellow.

What happens when white light is shone onto a coloured object?

The object reflects the colour(s) you see and absorbs the rest. This is why black objects get hotter in the sun than white objects. The white 'colour' reflects much more energ
In Science

What happens when red light is shone onto a wall?

Any object that has light shone on it would appear in a particular color because the said object absorbs all the other colors of the spectrum and reflects one single color.