Does an RF receiver constantly consume power?
Most receivers will consume power, when listening. (The only exception that I know of are RFID-type receivers.)

At the moment of writing, the specific receiver is not mentioned in the O.P.  Here are numbers for a Bluetooth Low Energy module (BLE113).


Transmit: 14.3 mA or 18.2 mA
Receive:  14.3 mA
Sleep mode: 0.4 uA (micro-Amps)


Cyclic sleep with wake-on-radio (WoR) is a common way of reducing the average power consumption and increasing the battery life.  This, however introduces latency.  If the radio is receiving for 0.1 sec every minute and not receiving for 59.9 sec, then the worst case latency will be ≈ 1 min.  As receive duty cycle decreases, the ampere-hours consumed in sleep mode will outweigh that during the active mode.

P.S.

The type and size of the battery is another major factor.  Achieving a 6 months battery life with a 10 mm coin cell would be harder than with a pack of alkaline AA cells.

Internal leakage in the battery itself (self-discharge) will also influence battery life.