Where not only the so-called TIPs, (by which is meant a certain series of reasonably popular power transistors in TO220 packages, designed by Texas Instruments) but also other devices such as 2N2222, LM386, and "bipolar transistors" and so on, are no longer to be used. Just because they might not be the best choice for switching loads controlled by an Arduino or similar.
So, I am not an EE, but it seems like what he's actually saying is that the MOSFET takes an order of magnitude less turn-on current and that it wasts an order of magnitude less power as heat. Is that true? And if so, why would you not want to save power? Are your driver transistors doubling as a heater?
A lot of the energy budget depends on the circumstances. When running on batteries, power draw is much more of a concern than when running on mains power. Similar with heating -- it may or may not be anything that needs to be attended to. Now having said that, there are several good reasons to use the MOSFET instead of the bipolar transistor, but they are not so overwhelmingly good that it makes sense to discard all kinds of bipolar transistors just because ot that.
The turn-on current for the MOSFET comes from charging the capacitor formed by the gate -- the instantaneous current is C dV/dt, in other words, the faster the transistor is asked to turn on, the larger, though briefer, will the current pulse required be. Once the transistor is turned on it doesn't require any current to stay on. There will however be another similar current spike, in the opposite direction, when the transistor is to be turned off and the gate capacitor is discharged. When the load is something like a motor, these time requirements won't be all that strict, so a controller is quite likely to be capable of driving the transistor. Now, without the necessary additional protective passive components (diode across the motor, maybe also a resistor and capacitor "snubber" circuit there as well; further diodes and resistors and capacitors around the MOSFET that serve to "eat" the energy coming from the motor being turned off) -- there is a nonzero risk that the load will turn on and stay on, having fried the MOSFET and maybe also the output circuitry of the microcontroller or Arduino...
In contrast, the bipolar transistor will require a steady base current for as long as the load is to stay powered, and they are more robust, less sensitive to surges and other influence of the load, and this does make it easier to make the circuit reliable, easier to make it work and easier to make it keep on working properly, The TIP120 and many of its relatives, being Darlington pairs, do have a fairly large current gain, so the required drive current is likely to be small enough for the microcontroller IO line to drive. But they have a voltage drop, and corresponding power loss, and are thus less optimal for driving motors. Which may or may not be a problem-- it will all depend on the actual application.
Thus, using a MOSFET for a switch has its advantages and its pitfalls like everything else -- and I don't have any objections to Tom Jennings recommending MOSFETs over bipolars for turning things on or off. What I do object to is the wholesale discarding of all bipolar transistors as if turning things on and off with the MOSFETs were the be-all and end-all of all electronics -- as we well know that is not so.