What are the positions of the Earth, Moon, and Sun at a time when there is the least difference between high and low tides?

Does this happen when there is a full moon?

'Neap' tides when there is the least difference between high and low tide, generally occur around the time of the first and third quarter moons. 'Spring' tides when there is the largest difference between high and low tide, generally occur around the time of the New and Full moons. The exact timing is complicated by your position on earth, the shape / size and depth of the ocean, coastlines also the moons distance from the earth and earths distance from the sun.

What are the positions of the Earth, Moon, and Sun at a time when there is the least difference between high and low tides?

TL;DR response: The simple answer to this question is twice per month, when the Moon is half full. A not so simple answer is roughly every 8.85 years, when the Moon is half full and the Moon is close to apogee. A full answer: Who knows? There are thousands of different frequencies involved, and their contributions change over time.

The greatest tidal variations (difference between high tide and low tide) occur when the tides induced by the Moon and the Sun are in phase. These are the spring tides, and this happens twice per month, roughly near full moon and new moon (but with some variation place to place). The smallest variations (which is what the question asks) occur when the tides induced by the Moon and Sun are out of phase. These are the neap tides, and this also happens twice per month, roughly when the Moon is half full, or first and third quarter.

The Moon's contributions to the tides is about twice that of the Sun (but this also varies from place to place). The Moon's orbit is eccentric, so tidal variation will be even greater when the Moon is full or new and the Moon is closest to the Earth. These are the perigean spring tides. The flip side of this effect, which is what the question asks, would be the neap tides that occur when the Moon is furthest from the Earth. These are the apogean neap tides. See Donald Olson (2012), "Perigean Spring Tides and Apogean Neap Tides in History," American Astronomical Society Meeting Abstracts# 219.

I've described three different frequency effects above. There are a huge number of other frequency considerations that affect the magnitudes of the tides. Modern tidal theory is the result of the combined efforts of William Thomson (Lord Kelvin), George Darwin, E.W. Brown, and A.T. Doodson. By Doodson's time, there were 388 different frequency components of the tides in use. My description above captures just three of them. Nowadays, people use over a thousand different frequencies to fully characterize tidal behaviors.

All of these effects are small compared to those that result in the spring tides and neap tides. The tidal variations (difference between high tide and low tide) that distinguish the spring tides from the neap tides can be huge, several feet in some places. The tidal variations that distinguish the perigean spring tides from the apogean spring tides are much, much smaller, on the order of inches. The other thousands of effects are smaller yet.