In most applications, the direction of the base current is always the same while the BJT is on (in forward-active, saturation or reverse-active modes): into the base in npn and out of the base in pnp BJTs. There are three typical exceptions I can think of.
1. Turn-off transients
If you connect a voltage source across a pn-junction diode in series with a resistor (such that the diode is forward biased), current flows in the diode in the usual direction. If you then suddenly turn off the voltage source, current flows in the diode in the opposite direction until the charge responsible for forward biasing the diode is removed. The time it takes for this to happen is called the storage delay time.
A BJT consists of two pn-junctions at least one of which (typically the base-emitter junction) is forward biased when the device is turned on. Much like in the case of a diode, the base current can temporarily reverse while the device is being turned off to eliminate the charge stored in the base, after which it decays to zero.
2. Reverse leakage current
If neither junction is forward biased (i.e. in cut-off), there can be a tiny leakage base current opposite to the usual direction. A typical example is when the base-emitter junction is off and the base-collector junction is reverse biased. Again, this is exactly analogous to the reverse leakage current in a diode.
3. Avalanche breakdown
If the reverse bias voltage of the base-collector junction is pushed past the breakdown voltage, avalanche multiplication results in a rapid rise in collector current. In the common-base configuration, this current goes out of the base of an npn BJT and into the base of a pnp BJT, i.e. in the reverse direction.