Is there a proper way to dispose of religious items, such as a broken rosary, a crucifix, a tattered bible? Both my sister-in-law and a co-worker refuse to throw such items out, as they feel just pitching them in the wastebasket is disrespectful. Any thoughts or suggestions?
Donate them to the church?
I am sure someone in need would be more than happy to recieve or repair the broken items.
I’ve always seen sacred objects (Bibles, icons, etc.) that were beyond repair either buried with somebody or burned and the ashes buried.
I wonder, too.
I make rosaries (of nylon cord, 8mm plastic beads, nickel plated medallion and crucifix) for my parish folks in general, and First Commmunicants in particular.
On occasion, I’ll find a crucifix I can’t use. The ring at the top, through which the thread passes, has broken off. But at 10 cents per cross, it isn’t a big deal.)
Up to now, I just put them aside to think about some other time. I guess I could just mail them back to the supplier (Our Lady’s Rosary Workers) and maybe they’ll take care of things. But that’s a cop out. I should come up with something better than that.
Talk to a priest. There is probably someone in the community who makes rosaries - both cord and metal link types. They should be happy to repair your rosary - or at the very least, to take the parts and recycle them.
This same priest might be able to tell you what to do with the other things you want to dispose of.
For what it’s worth, I’m taking this advice, too. 
Antiochus - I wouldn’t worry too much about the plastic crucifixes, considering they haven’t been blessed. To me, that’s like sacramental wine before the consecration - just plain old plastic. But once blessed, or used as the focus for prayers and meditation - I’d say donating them or sending them back would be appropriate.
StG
I think I said the crucifixes and medallions are nickel-plated, not plastic.
But I should hasten to add, StGermain, that I agree with the thrust of your post.
Broken rosaries-could you make jewelry out of them? I’ve seen jewelry made out of broken pieces of china, so why not rosaries?
I think that even as (I’ve heard) some Catholic churches have a spot of consecrated ground in which consecrated unused Hosts are buried, that ground might be used to bury such beyond-repair religious articles.
I had a worn-through scapular, a broken rosary & a tattered Child’s Bible (my own childhood Bible) and I had them all buried with my Dad when he passed away. We also had them include his permanently-stained coffee-cup & a pack of cigs. AND a little desk statue of a man with a big screw through him with the slogan “Work hard, do good & you’ll earn your just reward.”
The archaeologists who find him centuries from now will be SOOOO confused! 
WARNING: discursionary mode ON
there is a difference in disposing of sacred items as opposed to ones that are merely sacrementals. “sacred” would cover things like holy relics (of saints, the True Cross, etc.), consecrated hosts and wine, and maybe things like buildings consecrated as churches, chapels, cathedrals, etc. these are usually meant to be preserved if at all possible. things like dropped Hosts are picked up and (believe it or not) consumed anyway (assuming it is at all feasible). spilled Wine (post-consecration) by its nature is usually mopped up at best. in both instances, the area where they fell used to be “washed” with holy water afterward (this part may have fallen by the wayside). churches, etc. that will no longer be used as places of worship are de-consecrated, and may be re-used as regular buildings afterward or demolished.
sacrementals are supposed to be disposed of in a reverent manner (akin to the treatment of many sacred objects that have “passed their prime” throughout history and many religions. see “buried statuary from Acropolis”.) however, there is no particular sin or guilt associated with disposing of them or even necessarily treating them with less than full reverence. [hence, when Japan started a persecution of native Catholics, said converts could trample a crucifix when ordered to without grave spiritual repercussions, since a crucifix is only a sacremental (i.e., something meant to remind one of spirituality or aid in prayer and striving for holiness).] other such items are holy water, blessed oils for anointing, blessed palms (from Palm Sunday), and probably other minor blessed items like pictures, statues, bibles, etc. regarding organic sacrementals, i believe the preferred disposition is to return them reverently to the earth. oils and water may be poured onto the ground or into a hole. palms are normally burned (often the ashes are recycled for Ash Wednesday). other items could probably be buried, burned, or (for really mundane ones, like holy card pictures) probably even pitched into the trash, where they can rejoin the firmament at the local landfill.
and i have an extensive collection of rosaries, as well as doing repair work on them. check out like-minded people–sometimes they too are happy to take broken pieces, for repairing others or to try and fix themselves. (as ubiquitous as plain-black-bead rosaries appear to be, you have no IDEA of the variety of “plain black beads” that are out there!!)
In Judaism, any text containing the Divine Name (Torah scrolls, prayer books, etc.) and no longer usable is disposed of by burial. Texts referring to the Divine Name indirectly (such as this post) may be disposed of normally, as long as the text is handled with respect.
Religious articles, such as a mezuzah (doorpost fixture) without its scroll, candlesticks, and so forth – I have no idea. So I’ll be watching this thread to see what ideas people have.
ISTR it’s considered good practice to save the lulav (branches of palm, willow, and myrtle trees) as kindling to bake matzot (bread that tastes like cardboard).