Connecting devices with different logic levels
Anytime you exceed the limits listed in the datasheet you don't know what the device might do.  5 V is simply out of spec for the 3.3 V device, so it could fail, get hot, not work right, vanish into a greasy cloud of gray smoke, or whatever.

What actually happens in many ICs is that there are protection diodes from external pins to the internal power and ground nets.  By forcing a pin to a higher voltage than the chip's supply voltage, the diode from the pin to the power net will conduct.  This can cause currents in places there aren't intended to be currents during normal operation.  That can cause all kinds of off-spec behavior.  If this current exists while the power is coming up, it could even make the whole die go into SCR latching mode.

Of course the above was only one explanation of what happens in some ICs.  When the spec says the maximum pin voltage is 3.3 V, you should not apply more than 3.3 V and not assume you know what might happen if you violate this spec.

Some 3.3 V chips have "5 V tolerant" inputs.  In that case the chips contains additional circuitry so that 5 V on the pin won't hurt anything and will be interpreted as a normal digital high level.  For such inputs, it is fine to drive them directly from the output of 5 V logic.  But, this is only valid if the datasheet explicitly says it's OK.  Usually the valid voltage levels for digital inputs is well specified in the datasheet.  Take a look.