Having read the Wikipedia page on superfluids I'm still not sure if stuff like liquid helium at the lambda point actually have surface tension or not. Is superfluidity the same thing? And are there any liquids with no surface tension at room temperature?
Surface tension is not a property of materials but of interfaces between two (or more) materials. It is implicit in its definition that the interface separates two kinds of materials that behave differently (otherwise the interface would be just some imaginary surface inside the one material with no physical meaning) and so there must always be some surface tension that sustains the physical interface.
First of all, Marek is right that a surface tension exists only between two different materials (well, I would say between two different phases - for example water and ice). So let's rephrase the question as "Are there two phases with zero surface tension?" and elaborate a little on the answer.
The surface tension is the excess free energy (technically the 'grand potential') associated with the area of the interface between two phases. If the surface tension is positive (it always is), the system minimizes the free energy cost by minimizing the area of contact. This leads, for example, to the spherical shape of a water droplet in coexistence with water vapor. But if the surface tension were zero, we could deform the shape of the water droplet arbitrarily, with no free energy cost, so long as we didn't change its volume! We could even deform it to the point where, from the macroscopic point of view, the water and the water vapor seem to be perfectly mixed. But this actually contradicts the fact that there was phase separation in the first place, because phase separation is an indication that a uniform (single-phase) system can lower its total free energy by splitting into two phases.
So zero surface tension between two phases would be quite an unrealistic situation. There are some situations in physics which are somewhat like having zero surface tension. If you look at the surface tension between water and water vapor as the critical point is approached, the surface tension vanishes. But at the critical point itself there is no distinction between the two phases (and therefore no interface), so we can't say that there is zero surface tension. Another situation where it is tempting to say that surface tension has been made to vanish is when oil and water (which don't mix) are emulsified by adding a surfactant or emulsifier which lowers the surface tension. Eventually, rather than two coexisting phases, we have a single phase with a lot of internal structure (little droplets of oil in water or vice versa). But that is a single phase, not two coexisting phases!
So I don't think that any two coexisting phases have zero surface tension. In particular, liquid helium has a positive surface tension (with air I mean).