Where can I dispose of some large trash in Budapest?

Wait for the spring junk clearance days, there will be a huge pile of junk somewhere close to you with gypsies scavenging it and assaulting you if you touch any of it, just throw it on the pile.

Yeah that's one way - thanks! But I live in quite a small apartment, so I don't really have anywhere to hide it for now, so just wondering if there is anywhere I could take it before "trash day" comes around?

You could hire a bin or ask one of your neighbors where they may have a large container near by.
Down the st. from us there is a large hospital and church where some people put these items in the bins. They are out in the open on the st. and no one really seems to know who put them out. Often people put items like old clothing racks and small broken odd shaped pieces of furniture.
I would ask to make sure it was ok to use the bins though, don't want a ticket for making a mistake.
Otherwise once a year they have a city wide throw out, if you have a cellar you can store the chair until the next toss out.
Suppose you might be able to take it apart and put it in the recycling bin if it fits in.
Metal parts in the recycling and any soft materials in the regular bins?

I agree with Marilyn, taking it apart, or breaking it up, then put in appropriate local bins.

You can also self haul it to a land fill (they will charge you for the item). I have done that, or hired a truck to do it when I had a lot of junk to remove.

Or maybe take it to a used furniture store as a donation. What you call broken may be a weekend fixer project for someone else.

Thank you both for the replies, some great advice - thanks!

Your welcome, nothing is "too weird" to ask here, we have all been through the wringer one way or the other with trying to figure it out over here.
One day I was carrying down a broken clothing rack because we were going to take it over to the bins near us that that extra pieces.
My husband tends to dottel sometimes so I went down first with the rack, was about to go back and check on him so left the rack downstairs for a second, or rather tried to leave it, some women in our house, some lonely looking older lady( I'm 63 so she was old before her time) started going off being rude in Hungarian to me about leaving the rack downstairs. I told her off in my broken HU and said "wait a moment" in HU but her mouth wouldn't stop. My husband came down and we took the rack away but now I always give her a dirty look if I see her, what a freaky mean old B she is!
I have "turned Hungarian" in our house, hardly smile or say hello to anyone these days, learned the hard way to just put on the ugly face and then you are safe from rude remarks, weird seems the meaner one is the more respect you get over here...Not my normal style but whatever..
Now when I run across her she seems timid around me, makes me glad I must admit.

Haha, well I certainly don't want anything like that happening my Hungarian isn't remotely good enough to help me out of any awkward situation like that. Luckily I've not ran into too many rude people yet. Hopefully my neighbours are a little more helpful and understanding if I end up doing anything wrong.

Most people are more helpful. Just my bad luck , ran into the "weirdo" of the building, there's always at least one in any house.( Who knows, maybe for some people I may be the weirdo in the house! No, can't even speak to anyone really and do mind my own business unless they make it my business)
She was just being a "busy bee" that day. She should of minded her own business because we never even intended to leave the rack in the trash bins. Enemy for life now.
This was a long while back,since then other's in the house have left old sinks, used old toilets and cabinets in the common area for weeks on end before getting a bin to haul it away.
Sort of makes me think she was so mean because she knew I was not a Hungarian and she thought she could talk cr** to me. No, I don't speak Hungarian well but know when I am being bamboozled by the person. Just need to know about 30 words and how to use them and you can get by here.

Is it rubbish, or stuff that can be fixed? Stick it on Craigslist? Or find when the tip is open in your area. Stick it outside on the day and it will be soon taken.

There is a good site to resale items but it is in Hungarian.
Maybe someone can help you post as a freebie in the online add.
The site is Varta.HU

You mean Vatera.hu.