Almost zero, as they are basically capacitors with some slight dielectric absorption ( which is the light) but you can expect around 2-10nF per segment, depending on the size. Your display is essentially a very large segmented capacitor with a saving grace of having a moderate ESR as well due to the thin ITO layer needed to make it both optically clear and conductive on the front, though the rear contacts are generally a silver plate on the EL phosphor layer with an etched display layout on it.
Generally the front will be masked to create the digits, cost depends if it is front of the glass or done with a glaze frit mask on the ITO layer before the EL screen print, which gives a much better contrast between digits at a quite lower yield and higher cost.
The current is basically a capacitor with ESR of around 100R ( varies quite a bit) in series, though with a reed relay switch I would also add a series 1k resistor to each segment as well to keep switch current low, as you really cannot use any snubber across the switch elements.
Electronically an optical MOSFET switch is nearly perfect, just have to use one with 400VDC rating, and if it is polarised then a bridge rectifier with each one will be needed, but with both being SMD you could make them a lot smaller in form than a relay ever could hope to be.
Example is AQW280EHA at
http://docs-europe.electrocomponents.com/webdocs/1314/0900766b81314265.pdf which will do this without any problem with only a 5mA drive. DIP8 package as well is pretty small, available as through hole or SMD mount as well in the same size. You could fit them into the board space easily with room left over, and could even make some metal covers to emulate relays for them as well.
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