How close can you plant Willow trees (and other vigorous, damp loving species such as Poplars and Alders) to a house or drain?

The easy answer is "not close at all" to any drain, including modern drains, or old house. If there is a way in, willow will almost certainly find it.
Yes, we also have a friend who had an uncle whose dentist had a willow next to her well, and it was fine, but these exceptions only prove the rule.

A golden weeping willow is as beautiful and delicate in the breeze, but there is a monster lurking below.

Willows have to be the all-time root producing world record holders, and Alder and Poplar aren't far behind. Their tentacles will sniff out water and drainage pipes at long range.

Even if the roots cannot get in at first, there will be condensation on the pipe that encourages the roots to wrap themselves around the pipe like a sleeve. Perhaps the roots dowse for the water, who knows.
This should not an issue if the willow is squat and stable, but a tall tree that sways in the wind will tug at pipes wrapped in its roots, potentially working a gap loose for them to enter the pipe.
And if that tree falls over in a storm ... messy.

Without knowing for sure what their maximum reach is, I think it's wise to leave at least 10 metres between tree and drain: 15 metres feels better.

The company Water Willows, supplier of fine unrooted willow cuttings, gives these willow planting instructions:

"Do not plant willow close to buildings, walls, stone structures, drains & septic tanks where their roots may cause damage, as willow needs plenty of water and the roots actively seek it.
We suggest a planting distance of 1.5 times the maximum height that the plant will be allowed to grow. This applies to cuttings, whips, rods and trees, especially the more vigorous varieties."

Which brings us to a good argument that goes like this:

• Most tree roots extend underground only a bit further than the canopy's shadow above ground.
• A smaller canopy means (in theory) smaller roots.
• So by pollarding your willow to keep it small, it won't be a danger to drains. Happily, pollarding willow is a common garden practice.

That all sounds good, but this horror movie, available to you only here, shows you why your mileage might vary.

The Willow from the Black Lagoon

This poor little grey willow, Salix cinerea, which produces some of the most beautiful catkins, was only looking for a better life in someone's drainpipe.

It was 2-3 years old, pulled out by a friend of Rebecca, who works here and brought the creature in.

Those roots are well over 3 metres long, while the scrappy little plant is only 1.5 metres above ground.
The root mass is perfectly tubular and solid: a real tug of war to get it out, and another year would probably have seen it a metre longer and completely immovable.

So, if you see drains growing near your willows, it's probably better to get someone to do something about it.