OK. Assuming 20 degrees at the bottom of the mountain the top is going to be at about -9.4. If we assume a constant temperature drop then we cross the line of 0 degrees at about 2000 meters, so the last 1000 meters of mountain is going to be snow. I'm going to simplify some of the maths here and assume that all the snow is at an average temperature of about -4 degrees (as more snow will be lower down the mountain.
I'm also going to assume a uniform thickness of 5 meters and a perfectly conical mountain with 35 degree slopes (snow will likely start to slide down if it's much more). That gives us a volume of 525,000,000 m3 of snow.
Now this is where my assumptions start to get a bit hazy, since snow can be anywhere from 0.1g/m3 to 0.8g/m3. I'm going to assume about 0.5 to account for old compressed snow and new fallen snow (plus it makes the maths easier). That makes 2,625,000 kg of frozen water overall. We need 4,395,300,000 j to heat all the snow up to 0 degrees (remember my naive assumption about average temperature) and an extra 87,412,500,000 j to actually melt the ice once it's at 0 degrees (hooray for enthalpy of fusion), giving a total of roughly 91.8 Gj required to melt all the ice into water.
1 j/s is 1W. There are 86400 seconds in a day. So you need a power source of at least 10.6 MW directly (and constantly) applied to melt the snow in a day.
This raises a few issues, but also a nice possibility:
Not many natural (earthbound) things have that kind of power output. Even volcanic eruptions don't usually maintain that level of heat for long (though it has been known to happen the issue then is the volcanic eruption itself, not the water). The sun gives us 1.3 KW per meter, nowhere near enough to even heat the ice to 0 degrees in a day. Artificial means (like burying a nuclear reactor) can provide that much heat, but unless they're carefully regulated you can run into all manner of issues.
But what about geothermal energy?
The biggest geothermal plant in the world is currently the Geysers installation in the USA, with 22 plants and a total capacity of 1500MW. Divide that by 22 (again, naive, but I'm going with it) and you get 68MW. If a magma pool rose and started to heat subterranean water sources you could theoretically have a series of steam vents throughout the mountain venting superheated steam left right and centre. If the mountain is mostly scree underneath the snow (rather than solid rock) then this steam can be fairly efficiently routed to all of the snow, leading to a whole lot of melted water. Naturally some places would get more heat and thus melt faster, and some slower, and I can't even begin to predict the vagaries of that, but hopefully this gives you some idea of the numbers involved.
This would lead to 2625 tonnes of water starting to cascade down the mountain. Unless the rivers off the mountain can cope with over a hundred tonnes of water an hour or your village is on high ground you can expect that they are not going to fare too well.
Quick note: My maths was done hastily and naively. If anyone spots a mistake/wants to improve it, feel free