What would happen if a jet liner flew with its undercarriage down? Would it merely slow it down - or would it actually endanger the flight?

Simply flying around with the gear down would not endanger the flight. (Whether of not it slowed down would depend on the amount of thrust applied by the crew. They could literally blow the gear doors off!)

The main concern one would have structurally is for the landing gear doors, as there is a design speed (velocity = "V" ) for them to be open. If the aircraft accelerates past that, they could be torn away. There is also a design speed limit for the gear to be extended. If that speed is exceeded, then undue stress could occur to the gear.

"Vle" - the maximum landing gear extended speed. The maximum speed at which the aircraft may be flown with the landing gear extended. VLE is always higher than Vlo

"VLO" - the maximum landing gear operating speed. The maximum speed at which the aircraft may be flying while raising or lowering the gear. VLO is always lower than VLE

There was an incident many years ago in TWA where the aircraft started began to oscillate at altitude. In an effort to stabilize the aircraft, the Captain lowered the gear at a speed far in excess of max speed for gear extension ("Vle"). It worked and they managed to land safely, even though the gear was quite literally bent backwards several inches!

http://aviation-safety.net/database/record.php?id=19790404-0&lang=en

As mentioned, another factor is the drag. More drag increases the fuel burn. Intentionally flying with the gear down may be done by the crew in the event they need to burn fuel quickly for a sudden return to the airport so as make their maximum landing weight limit if the aircraft is not capable of dumping fuel. When airliners talke off, they are often above their max certified landing weight as they are fueled up for a journey. Thus, a sudden need to return to the airport would require them to burn fuel off as quickly as possible.

http://en.wikipedia.org/wiki/MTOW

(You may remember the JetBlue Airbus that flew around Los Angeles for 3 hours last year trying to burn fuel as they needed to return for landing after a gear malfunction.)

It would slow down due to aerodynamic drag.

Additionally, it may pitch as the drag is down low, possibly below the center of mass. Depending on the degree of this pitch, it may actually endanger the flight if the aircraft speed is excessive.
However the plane IS designed to fly at lower speeds with the gear down for takeoffs and landings and any competent pilot will know the aircraft well enough that it will never actually be an issue in the first place.

I think its reasonable to include the landing part as part of the flight. I should imagine that quite a heavy frost would form on the gear and cause the plane to crash in some manner on landing?

It depends on what type of Jet liner it is. The larger ones like the 777,747, or the A-380 it could easily endanger the flight because of the speed they have to move at to keep from stalling can easily exceed 200 mph for a fully loaded A-380 and the cruising speed its 560 mph, at these speed the landing gear can be damaged due to the force and friction of the air.
On smaller planes it would be much less threatening as the speeds that they fly at are much lower. The force and friction most likely would be within the operating range of the landing gear. The effect of lowering the gear would have on the aircraft would be to cause massive amounts of drag, and slow the aircraft down.

No danger whatsoever as long as speed restrictions are observed. A few years ago, I jump-seated on a major airline flight which had been over-fueled, and we flew the last 250 miles with the gear down to burn off fuel so as to get down to landing weight.

It depends on what type of Jet liner it is. The larger ones like the 777,747, or the A-380 it could easily endanger the flight because of the speed they have to move at to keep from stalling can easily exceed 200 mph for a fully loaded A-380 and the cruising speed its 560 mph, at these speed the landing gear can be damaged due to the force and friction of the air.
On smaller planes it would be much less threatening as the speeds that they fly at are much lower. The force and friction most likely would be within the operating range of the landing gear. The effect of lowering the gear would have on the aircraft would be to cause massive amounts of drag, and slow the aircraft down.

  I very much doubt if it would pose any danger to the flight.  It would increase air resistance, which would very likely have the effects of slowing the flight down somewhat; and of causing it to burn more fuel to cover the same distance.