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Instructions on resin casting with silicone moulds frequently say to use a long necked lighter on the surface of two part epoxy casting resin to remove bubbles.
If I am in an environment where naked flames are prohibited can I use a hair dryer or another similar heat source to achieve a similar effect or does it need to be a flame?
A hair dryer is actually a bad way to deal with bubbles. It blows too much air, too little heat, and can blow dust onto the resin.
Start by minimizing bubbles
There are many techniques to minimize bubbles in the first place. That would really be a good subject for a separate question, so I won't get into that tangent here. But that's the first step --minimize the bubbles you will need to deal with. Avoiding as many as you can is better than trying to get rid of them.
Kinds of bubbles
There are many kinds and sources of bubbles. If you own a vacuum chamber, you can eliminate one source of bubbles by extracting them from the resin before you pour it. If you own a pressure pot and have a small mold that will fit inside, you can shrink tiny bubbles that have trouble rising to the top.
Otherwise, as a practical matter, you can deal with just a few large bubbles deep in the resin, or bubbles that rise to the surface. If the resin is viscous or fast-setting (hint: don't use fast-setting resin for molding or coating), bubbles may have trouble rising to the surface. Those you can encourage out with a pin or toothpick. Surface heating isn't an effective way to get those out.
So it's surface bubbles that you're trying to pop.
Ideal solution
The best thing for that is a naked flame; it's high heat that will instantly pop the bubbles without significantly affecting the cure, with very little air movement and no dust. In fact, it can burn off dust, leaving much less visible residue. You can use a small butane lighter rather than a blow torch. You pass it close and quickly over the bubbles. However, if you must work where naked flames are prohibited, even that isn't an option.
Alternative solution
The next best solution will be a source of high heat with the least possible air movement. A wind can affect the resin surface. It can blow resin out of the mold or cause excessive resin to spill off a flat surface you're trying to coat. A wind causes unevenness and ripples in the surface. If you combine that with heat and overdo it, you can end up with a surface that isn't mirror-flat.
The closest thing to this requirement is a heat gun. The best type for this is an embossing heat tool, the kind used for working with embossing powders. They get hot with minimal air flow because that would blow away the embossing powder. If you don't have one of those, a regular heat gun is the next best thing. If it has a high and low setting, and the high setting also blows a lot more air, use the low setting. If the high setting is just hotter, use the high setting.
You will need to get the feel for speed and distance. You need to hold the heat gun far enough away so you aren't blowing a strong wind on the surface (around 6" for a hot blower with low air volume). You don't want to keep it in one area until bubbles pop; it should be hot enough at the surface that bubbles pop quickly as you pass over them.
Here's a quick video from a resin supplier on doing it with a regular heat gun: https://www.youtube.com/watch?v=oZckiG0CQz0
Edit: I'll add addressing your comment about using a soldering iron or burning tool (a great question). That's a bad idea for two reasons: it's bad for you, the resin, and the tool, and it isn't effective.
Bad for you, the resin, and the tool
Resin that touches the tool will cook, putting off bad fumes and smells that can range from obnoxious to hazardous. While the tool is touching the resin, it will start to cure the resin in that spot, so the spot may not end up mirror flat, and worse, may discolor or start to boil and create even more bubbles. It will also be a mess to clean off the tool.
Ineffective
The purpose of the heat source is to cause bubbles of all sizes and quantities to pop without having to touch the surface. The kind of heat that's useful for this is high heat that causes rapid expansion of the air in the bubble so the bubble pops before the resin surface gets hot.
A soldering iron or burning tool can get very hot, but they don't radiate enough of that heat to be effective at popping bubbles by just waving it near the surface. Air is an insulator, and the tool has little radiating surface, so not much heat moves from the tool to nearby objects. A heat gun is designed to heat air, and uses air to transfer the heat by convection.
A soldering iron or burning tool would offer no benefits over popping large bubbles with a pin. The problem is that there typically aren't just a couple of surface bubbles, and some surface bubbles are tiny. It usually isn't practical to try to deal with them individually, and you can't pop tiny bubbles by poking them.
