As I understand the physics of strings the fundamental and the harmonics are in tune as long as the diameter of string is small compared to the length. As the diameter increases the harmonics become sharp relative to the fundamental due to the string becoming more like a rod which makes it less elastic to smaller waves which have higher frequencies. This raises the pitch of the harmonics. This is why a piano tuner uses a stretch tuning on piano. The low strings harmonics are sharp because they have a larger diameter. Since you hear the harmonics more than the fundamental in the lowest notes of the piano the tuner sharpens the higher strings to be more in tune with them.

The ideal piano would have very thin strings with equal tensions but different lengths. To achieve this ideal the length of the lower frequency strings would have to be so long as to make the whole instrument huge. To overcome this the designer of the piano uses thicker strings for the lower notes. This is because heavier strings vibrate more slowly than thin strings.

With a thin string, the overtones or harmonics will be at multiples of the fundamental frequency. With thicker strings, the force required to bend the string as it vibrates at high overtones increases. When this required force is much less than the tension in the string there is little effect. When the force starts to aproach the tension in the string, the wave velocity of the string increases and the frequency increases. Hence the overtones will be sharp and the undertones will be slightly flat.

To an extent this can be overcome by using wound strings. Because these strings are thicker their fundamental frequency at a given tension and length will be lower. Because the bulk of the mechanical resistance to bending is in the thin core of the string, the overtones will not be as sharp.

A piano tuner tries to get the overtones to a frequency as clos as possible to the correct value. This results in slight flattening of the fundmental. Because of this tuning a piano is a compromise.

The longer the strings of a piano can be, especially the lower notes, the smaller this effect and the better the piano will sound. This is why a large grand piano is better than a baby grand and much better than an upright.

Excellent though the above answers are, it's probably not just due to the strings themselves.

The frame of a piano, or body of an acoustic guitar, are constructed as lightly as possible provided they are strong enough to support the string tension. Partly this is to minimise weight and materials, but mostly to maximise the transfer of energy to the body of the instrument, which acts as a sounding board. If a piano were made of girders, the notes would persist too long, causing discords with following notes, and the sound volume would be minimal. (Consider an electric solid-bodied guitar without an amplifier.)

When I tune my Spanish guitar starting with the lower strings, winding up the top strings bends the body enough to relax the lower strings and make them flat. I have to work round the whole set at least three times to overcome this.

As a string is plucked or hammered, the tension in it increases slightly because it is extended sideways and thus is stretched tighter. (Consider how a guitarist 'bends' a string to get a note up by a whole semitone.) For a high note, this would make the note slightly higher. For a low note with a wound outer skin, it would be more likely to bend the frame or body temporarily, and thus flatten the latter part of the note. Sounding a number of adjacent notes would presumably have a cumulative effect on the frame and flatten the entire chord.