Implications of "旧的不去，新的不来"
Can this be used for people hold onto broken items/trash? 


Yes. This is the most common usage and safe to use.

Can this be used for getting over failed relationships? 


Yes but don't use it if there is still chance for the relationship to be fixed or if the other party is also your friend. Because 旧的 (old thing) is more or less derogatory.

Can this be used for social changes like the acceptance of homosexuality?


Yes it can be used for social changes, in this case not a as consolation but a neutral sentiment. Please note the literal meaning of the phrase must meet i.e. there has to be an old thing being abolished and a new one being made. The emergence of new thing alone, gay marriage for example, is not enough to apply the phrase.

Are there any particular uses that it implies first and foremost?


As mentioned above, the first implication and most common usage is to console someone who can't get over broken things. Sometimes people use it on failed relationship or even dead pet, which is not always appropriate due to the derogatory sense of 旧的 (especially when put aside with 新的). To be safe, use this phrase only when you're certain the old thing is not worth holding onto and the new thing is better than the old.

There is a related but generalized usage, that is on new year's eve people say 旧的不去，新的不来 as a wishing message. In this usage 旧的 refers to all misfortunes and unnecessary concerns occurred in the past year and 新的 refers to all good things to come.

Another usage is to express a neutral comment as an observer of a change like in the social change example above. It carries little to none personal opinion, mostly a statement of the fact. This usage can be sometimes found in news or social chitchat but not really common in daily conversation or serious discussion. As a matter of fact when someone's asking your opinion about some change, you can use this phrase as hand-waving - it sounds like an opinion but it's just a recap of the fact.