Can't see your photos..says they've been moved. Anyway, it's a bit early for poison ivy to leaf out while honeysuckle is beginning to leaf out (here)already. Is this a fence location where you/spouse and children are likely to be exposed to poison ivy (if that's what it is)? If not and since this is a rental and the vine and leaves are screening out a bad view..leave it. If you're concerned, then wait until it leafs out and check google for photos of poison ivy for identification.

"leaflets three, let it be". Looks pretty suspicious to me but have someone else confirm it for you. If it is, put on the gloves, long sleeves, hat, and have the "Tech Nu" lotion standing by (great stuff to get the oil off your skin so it doesn't spread). Might also want to apply round-up to those plants if confirmed, but don't think because it looks dead it can't spread that oil, even the hairy roots and dead looking vines ... it can, and never burn the plants you clean out.

By the way, every part of the plant carries the poison oil ... even after 2-4 years, they say. There's many links for this if you Google it, but here is one I ran across.

Here is a link that might be useful: mash here

Poison ivy in full flower in that first picture. And yes, poison ivy has beautiful fall colors!

Those are all poison ivy. And big healthy established vines by the look of those blooms! What a job! If it were me, i'd don heavy duty protective gear in 2-3 layers(preferably an outer layer you can toss afterwards), and start cutting away as much as i could while it's dormant and fairly easy to work around. That won't kill it- but when it reprouts at the ground where you cut it off you can hit that with roundup and not damage other stuff. If someone in your family is more resistant to the oils, let them tackle the job.

When you're cutting big vines especially, do it slowly so it doesn't splash/squirt sap around. Probably a good idea to wipe down your pruners as they get wet, too. Not sure how 'juicy' it'll be this time of year- i know at other times, the big vines are pretty sappy. It's going to be awfully hard to spray it thoroughly and not get anything else if you don't trim some of it down. You're taking some of the plant's system and weakening it some by trimming. Plus, it takes more spray if you leave it as it is than if you trim. Best of luck!

As others have said- be very careful, and wash with technu really well afterwards. If you do end up with a rash, zanafel works pretty well at reducing the time you have it (it's expensive, though). There are a couple of prescription creams that will help tremendously in drying it up as well, and they aren't commonly known. The steroids and shots do work, but i don't like dealing with either one. The dormant vines can still give you a whopper of a rash, and the sap is the worst.

If it were me, and this rental had it before you moved in, i'd ask the landlord if they'd knock off a little rent or pay for the weed killers for you. It's a nasty job, and you should get some compensation.