The Moon's position affects tides. So, is there any possibility for man-made satellite positions to also affect tides?
Note: this is a 'Fermi estimate' answer, so I'm going to round off and ignore minor effects.
Tides are caused by gravity, and gravity is a really weak force. The mass of the Moon is 7 x 1022 kg, and it causes tides where the difference between water levels is on the order of 10 m.
Satellites are on the order of 103 kg, and they're 103 times closer, so the tidal effect caused by the average satellite is 1013 times smaller, and that's too small to be measurable.
Another factor is the number of satellites: 1000 sats in orbits evenly distributed around Earth will cancel out each other's tidal effects.
Everything that Hobbes says in his answer is correct, but doesn't seem to me to give a clear answer, which is YES they do affect tides, though not to a measurable extent.
When they were launched, that had a tiny effect on the earths rotation, though not particularly relevant here.
Every single one has exactly the same effect as the moon does, but to a far smaller extent, due to the tiny mass. The totality of artificial satellites, including the ISS, are NOT uniformly distributed, so they do not entirely cancel each other out.
Let's do one small calculation.
The mass of the Moon ~ 7x10^22 kg, roughly 4x10^8 m away.
Mass of a really large satellite (ISS) ~ 5x10^6 kg roughly 4x10^5 m away.
The magnitude of the tide will be proportional to the mass of the orbiting object, and to first order, proportional to the cube of the distance.
For the Moon, M/(r^3) is roughly 0.001
For the ISS, M/(r^3) is roughly 8x10^-11
So, the ISS raises a tide that is around Seven orders (~1x10^7) of magnitude weaker than that of the Moon.