This is easily fixable, and shouldn't be too expensive although the cost of materials isn't negligible.
All you have to do is remove the softly cured excess oil from the surface, which should take all of the crud with it. After this, a quick rinse down with solvent and/or soapy water and the surface could look virtually brand new.
We actually have an existing Q&A on the same topic, How to solve the behandla problem -- removing or avoiding sticky oil residue although this is a little different due to the timeframe. Since the oil in this case has had longer to cure and subsequently get dirty the approach needs to be altered and this is why this isn't a simple duplicate Question.
IKEA benchtop (Karlby) which I coated with Behandla (danish oil) that comes with it and have obviously put too much on.
I suspected immediately on reading the Question title that this would be a problem with the use of Behandla.
Note that unless they've changed the formulation for Behandla it's actually primarily linseed oil, not a version of "Danish oil", although both can leave a gummy/sticky surface if all excess is not removed after application1.
How to fix
Although you indicate you've tried acetone and it didn't do much it should actually work for this. But you have to use significant quantities and it should be used along with a pad of steel wool or a nylon abrasive pad to scrub the surface, not just a wiping cloth or a pad of paper towel. So if you don't skimp on the acetone or effort it should get the job done2.
However, it would be more effective to use a stronger solvent that evaporates more slowly or a commercial finish stripper, and it's the latter I would suggest you use.
Strippers vary is speed and safety level. The fastest and generally most effective type (methylene chloride) is unfortunately the most hazardous to use and I can't recommend its use in an enclosed space by someone unfamiliar with it3. There are now safer strippers to pick from which for work indoors I think make a lot of sense. I don't want to promote a specific product but 'green' strippers which are often a gel consistency with a citrus smell should work perfectly for this.
To use you apply a generous coat, then leave it alone to do its work for a given time (15 minutes may be enough here but if it takes longer it takes longer). Then carefully scrape the stripper along with the softened finish from the wood. If necessary you repeat the process on any stubborn spots. Tip: expired credit cards or store cards make great scrapers.
After stripping you clean the surface down to remove remaining traces of stripper or finish. Pros often use lacquer thinner for this but since you have acetone I'd suggest you use that, if there's enough remaining. And after this you may also want to rinse with soapy water, wiping dry when you're done.
Then let the surface dry out thoroughly, ideally overnight, before refinishing.
Safety note: I would wear robust gloves throughout for this kind of work, no matter how safe the stripper says it is on the labelling! And eye protection should be considered mandatory.
Then you have to decide what, if anything, you want to do to treat the countertop since stripping will likely also remove all of the original finish if there was one (see note * bottom).
You could use the Behandla applied correctly this time :-) although I don't think it provides enough protection for a working surface in a kitchen myself.
Actually using "Danish oil" (usually a varnish and oil blend) is an option, but I think it's better to go straight to varnish myself. It can be just as easy to apply, in a wipe-on/wipe-off manner, and it's tougher, more durable and more water-resistant. See more on wiping varnish in this Question.
Please note that surfaces treated with virtually any oil-based finishing product aren't considered cured for perhaps a full month after the last coat is applied.
Our handyman has simply said to replace the entire bench but we can't afford that!!
If I might offer an opinion on that, it sounds like they simply don't want to face the labour of stripping the gunk off the surface! Which I understand, it is a fair bit of work. It might take a concerted hour or possibly two, after which your hand and arm will be tired but it's hardly what I'd call backbreaking labour. Even an unfit sedentary person can do this kind of thing...... I speak from experience :-)
However, I hate to say this because of the waste but given the cost of labour on top of the materials needed for this it is possible that complete replacement is actually the most cost-effective solution here.
At time of writing in the US a Karlby countertop is $169 without the Ikea Family discount. I don't know what the cost of labour would be to you but factor in approximately two hours to the cost of materials. The materials list might be:
- the stripper
- possibly more acetone, or lacquer thinner
- steel wool or nylon abrasive pads
- cleaning rags (old t-shirts or threadbare sheets are an option if you have enough)
- possibly gloves and safety specs
- tape and plastic sheeting/bin liners to protect adjacent surfaces
....it begins to add up.
In addition, note the curing time given above. While finished surfaces can be used lightly after allowing a few days to dry, the finish is not at full toughness and won't provide its full protection until it is fully cured. Surfaces should not be washed down during this curing process or even wiped with a damp cloth, which obviously may present a difficulty in a rental environment.
1 When you're finishing wood with an oil or oil blend the surface should actually feel quite dry when you're done wiping away the excess. I think the instructions on all products should be much clearer on this point, although the Ikea product does say "Wipe off any unabsorbed oil with a soft cloth."
2 Unfortunately acetone evaporates very quickly, which constantly fights your efforts to get it to soften a gummy finish. This is why straight acetone is not generally used as a stripper, even though it is very effective at softening a range of finishes.
3 Although perfectly safe if used correctly (it's the most widely used stripping chemical in industry and has been freely sold to the public for decades) I want to be very clear on the potential risk here for the benefit of future readers — incautious one-time use of methylene chloride strippers can result in death.
*The Ikea site for my country states that "For quick installation and easy maintenance the [Karlby] worktop is pre-treated with hard wax oil." If that applies universally you don't have to use anything on your Karlby worktops initially, although the surface will require regular topping up.