Can a laptop battery explode while using Linux?

Laptop batteries typically have onboard firmware to control safe charging & discharging of the battery, report battery charge level to the OS, and prevent thermal runaway, which is what will cause an Li-ion battery to explode (or more accurately, catch fire). Most modern ones also contain mechanical failsafes to prevent such fires & explosions.

This firmware is stored on the battery, separate from the OS. While it can be updated from the OS (although this depends on the battery & laptop), it's not something that is altered when installing a new OS or something that is typically ever tampered with unless done so by the user running a battery firmware update.

The only thing changing OS will affect is the load on the system & the hardware drivers used, not the safety features of the battery. Load on the system in and of itself will not normally cause issues with the battery other than faster discharging.

Interestingly, according to this forbes article, there was actually a vulnerability in Apple laptops (running OSX, not Linux) that could do nasty things to the firmware on the batteries - perhaps your Dad has read something like that which is why he seems to think the OS can do this?

(It's more than likely been fixed since 2011 when the article was written).

EDIT - in conclusion, aside from possible attack vectors for battery firmware hacks, the choice of OS alone cannot cause a battery to explode.

I think all batteries can explode. The question is if Linux (an operating system using the Linux kernel) will generate more heat or not. With good use of the fan (equally good use of the fan as other operating systems might employ), the cooling should be equal or better, thus resulting in a similar (or perhaps even reduced) risk of battery explosion.

There is also a matter of graphic cards and fans. Graphics cards may generate a lot of heat, and the fans may be managed by the graphic card drivers (kernel modules). These must also run properly for the system to be as cool as possible.

The 2013 Retina MacBook Pro uses either an Intel Iris Graphics 6100 graphics card, an Intel Iris Pro Graphics or an Intel Iris Pro Graphics in combination with an AMD Radeon R9 card. The open source Intel drivers are rock solid and should not cause any problems. The AMD drivers are more questionable, depending on which one you choose to use. If you use the Catalyst driver, the fans are supposedly both silent and efficiently managed (but there may be other issues, it has a bad rep). The open source radeonsi driver had a problem where they made too much noise, all up until early 2015, which is now fixed. Setting the fans to maximum speed at the early stages of graphics card driver development is common, and this only increases the cooling. This means that even drivers that are under development should cool the system adequately, and not increase the risk of an immediate battery explosion.

I'm pretty sure that an OS has no capabilities to destroy a battery (be it Linux OS, Windows, etc.). A malfunctioning program (and not Linux by itself) can overload a laptop/PC up to the point that, for example, it can enter CPU thermal protection.

But a fresh Linux install will not damage any hardware (if it does, than something is terribly wrong with your hardware).