Are there any airliners capable of vertical climb with engine thrust only by pointing the nose up? So, sort of inspired by this question, I was wondering if there were any passenger airliners with enough thrust to fly straight up, supported purely through the thrust of their engines rather than the aerodynamic lift generated by their wings - so that once it's flying along, it can just point its nose at the sky and go straight up; if you think about the launch of the Space Shuttle, you’ve got the right general idea. According to some of my comments, this is apparently called a "zoom climb". Note that I'm not asking about VTOL, since some aircraft that are capable of this may not be capable of VTOL due to a lack of thrust vectoring.

I know that some military fighter aircraft are capable of this, but based on some research with Wikipedia, it appears that the Boeing 747, Boeing 787, and the Airbus A380 can't - though the 787 would almost be able to do it if it had four engines rather than two.

Are there any passenger aircraft that are capable of doing this? Obviously they wouldn't do so while passengers were on board, but I'm just interested in the ratio between Operating Empty Weight and Maximum Thrust.
No, there are few planes that can do this at all. In order to be able to climb out straight up you need a thrust-to-weight ratio greater than 1. In other words you need enough thrust to propel the full weight of the aircraft up without the wings generating lift. 

Wiki provides a brief list of aircraft and associated thrust to weight ratios, the Concorde comes in on top at 0.372 with full afterburner but this is at max weight. Although at its empty weight it was pushing a 0.877 ratio. Even running on fumes would not have done it at full afterburner in the Concorde. For comparison the 757-33 has a ratio of 0.6 at Operational Empty Weight assuming a maximum thrust of 42,600 lbf per engine.