I personally think this is one of the most thematically satisfying and versatile weapons in the current card pool.
The paying 1 to boost 2 is an incredibly good rate of return compared to the likes of Hard Knocks or Physical Training (the only other pay to boost options in the game just now). In low characters such as Wendy it makes occasional combat quite viable even if you dont want to go all in for the + damage. A maximum of +6 should be enough to land a blow of some sorts on most enemies irrespective of which investigator your playing. In a combat focused character it may enable harder difficulties, or devastating late game combos with Double or Nothing and Vicious Blow. Note that Zoey's passive bonus resource upon engagement can be spent immediately on the axe if needed. She can also choose not to take it if you dont want to go above 3 resources to keep that + damage option open.
By combining a damage card and pay to boost ability, Fire Axe sets up much quicker and much cheaper than decks aiming to use Hard Knocks or Physical Training with a weapon (likely Machete) to maintain both high and bonus damage. The tempo boost from this is not to be understated here. In the case of Machete and Hard Knocks, you need to find 2 cards, pay 2 actions and 5 resources compared to 1 card, 1 resource and 1 action with the axe
To compensate for these fairly hefty advantages, Fire Axe requires a little working around if you want to reliably be able to use that +damage abiliy. Firstly it is important to be aware of your cash flow. Be careful not to play your Emergency Cache or similar until you actually need to spend it. in cheap enough decks you might not even need to bring a cashe at all.
Second or subsequent swings with bonuses to both accuracy and damage can be obtained through committing cards for icons on your second attack, in particular Overpower, Unexpected Courage, or the in faction Oops!. Alternatively, as pilgrim highlights, Forbidden Knowledge lets you take another resource between swings giving you a second attack at +2 and +1 damage.
Comparisons with competitors:
Machete is arguably the closest comparison for a weapon you could potentially plan to reuse indefinitely. The consistent + 1 damage is only dependent on the enemy being engaged with you and being the only enemy engaged with you (not hard to achieve by any standard) and it attackes with a passive +1. This may not be enough alone in low characters, but it probably is in combat heavy characters at least for regular encounter deck opponents. In exchange it is more expensive, a different faction and arguably needs support cards to hit boss type enemies (rather than resources as the axe does). That said, by the time a boss has materialised you would expect to have seen your combo pieces.
Baseball Bat offers a consistent + damage, and effectively saves you 1 resource per attack where you needed to boost, but will eventually break and similar to Machete needs support to go higher on bonuses should the need arise. Arguably its biggest achilles heel though is the two hand slots requirement, which can be very restrictive in non-arcane focused investigators, particularly by hampering the option to use key investigation cards like Magnifying Glass or Flashlight. Or indeed carrying a back up weapon.
Comparisons aside it is worth remembering that if you do plan on using weapons, you may well want 4 or 5 cards in the deck to be likely to draw one early. Certainly 2x Machete and 2x Fire Axe together isnt a bad call for a melee build. Late game you will probably have the one you need for your boss fights and early game frankly they are similar enough to be substitutes while you get rolling.