To summarize the two major arguments against building more rental housing:
1. Vancouver is over-crowded already. High housing costs are just the price of living in Vancouver, and it's actually a good thing because it keeps newcomers out. (Example: Colleen Hardwick.) This is the "we're full" argument.
The counter-argument is that high housing costs make us all poorer, pushing out younger and lower-income people especially. Renters have to deal with low vacancies, high rents, and insecurity - if they lose their housing, they're going to have to pay 20% more overnight, or move further out. First-time homebuyers are having to borrow enormous amounts of money. Even for long-time homeowners it's not great: they have massive gains on paper, but they don't benefit from them at all unless they sell and move. They feel like they're getting pushed out as well. Eventually all their neighbours will be replaced with rich people.
We can have a city with lots of housing and a range of people of different ages and incomes; or we can have a city with scarce, expensive housing where only rich people can really afford to live here, even they have to pay a high price, and everyone else finds it increasingly difficult to hang on.
2. A completely different argument: building more market rental housing isn't going to help with the affordability problem. What we need to do is build more rentals that are affordable for lower-income renters. (Example: Jean Swanson.)
The counter-argument is that blocking more market rental housing doesn't help to provide any more affordable rental housing, and makes the problem worse. We have lots of job openings in Vancouver, and as newcomers arrive to fill high-paid jobs, if there isn't any market rental housing available, they'll end up in the older, cheaper rentals, pushing lower-income renters out.
(There's plenty of recent studies showing that building more market rentals does in fact help with affordability, through "vacancy chains," but they tend to get dismissed as "vacancy trickle-down theory.")
In addition, the current Streamlining Rental policy being considered by council makes affordable rental projects easier and faster (not just market rental projects): in C-2 zones, they can build more rentals on the same amount of land, without having to go through the 12-18 month process of rezoning.