Does a capacitor connected directly to a battery consume any energy?

In steady state (after a long time) an ideal capacitor does not draw significant current from a battery. A real capacitor will draw some small leakage current. The amount of leakage current will depend on the type of the capacitor, electrolytics will have higher leakage than films and ceramics.

Leakage current will drain the battery, most likely not that significantly compared to the internal self-discharge of the battery.

An aluminum electrolytic might leak 100nA long term, which is not much compared to the self-discharge of even a button cell. The guaranteed maximum of a typical e-cap of this size is 0.002CV or 400nA (whichever is greater) after 3 minutes. Most parts will beat that significantly. Some SMD parts are not nearly as good.

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Your second question was whether this safe to make. Generally, yes, however there are almost always exceptions in engineering. If your 3V battery has a large current capacity (perhaps an unprotected 18650 Li cell) and your capacitor is something like a 6.3V tantalum capacitor there is a significant risk of an 'ignition' event upon connecting the capacitor to the battery (picture flames shooting out, a bright light and some noxious fumes). The risk can be considerably reduced by adding some series resistance of some tens of ohms.

An ideal capacitor would be open circuit to DC, so no current would flow, and no energy would be consumed after the capacitor is fully charged.

However, real capacitors do have some small leakage current, so, in Real Life, energy would be consumed from the battery very slowly after the initial charging.