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Tool Expertise At 6th level, your proficiency bonus is now doubled for any ability check you make that uses your proficiency with a tool. (Artificer)

Fire Rune This rune's magic channels the masterful craftsmanship of great smiths. While wearing or carrying an object inscribed with this rune, your proficiency bonus is doubled for any ability check you make that uses your proficiency with a tool. (Rune Knight)

I'm multi-classing and have both artificer and rune knight subclasses.

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    \$\begingroup\$ The answer on rpg.stackexchange.com/questions/44243/… covers this question. It's likely this question will be marked as a duplicate but feel free to leave it up as a way for others to find the answer. \$\endgroup\$
    – anon
    Jun 20, 2022 at 9:07
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    \$\begingroup\$ I've marked your question as a dupe (though not of the question suggested by others) because I think it has been asked before. This isn't a bad thing as this question might help others find that one. If that question doesn't answer it for you, please clarify for us how and why, and we'll see what needs to be done to help you. \$\endgroup\$
    – Someone_Evil
    Jun 20, 2022 at 9:15

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Doubling proficiency bonuses do not stack

See page 171 PHB:

Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue's Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.

In your case, both abilities multiply the proficiency bonus, but the rules says that in such a case, you only multiply it once. You will not quadruple your bonus.

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