If he took his money our combined with mine we could buy a house with no mortgage. However he is saying this is unfair because then his siblings mortgage would increase.
Of course their mortgage will increase - because they will be buying a whole house (and bigger than their existing one, by the sounds of it) as opposed to the half a (smaller) house they're buying at the moment. The new house will eventually be theirs outright, whereas they will only ever own half of their current home, unless they buy your DP out. Surely they can understand that buying half of something costs less (generally 50% as much) than buying the whole thing?!
I can understand how the investment arrangement might have started in the beginning, but wouldn't they have expected to split the rent from their tenant? Thus, when sibling becomes the tenant, they (theoretically) pay themselves half of the rent and pay the other half to the other 50% owner/investor. An alternative (probably a legal minefield, though) might have been an agreement that the split in ownership equity would be adjusted every month/year in favour of your DP as a trade-off for the non-payment of upfront rent.
The sibling (rightly) seems to see their half equity as a very valuable commodity but somehow (very wrongly) sees your DP's equal equity as a little something like a spare lawn mower or telly to be borrowed indefinitely.
This is madness. If they can't or don't want to get a mortgage on a whole house, they can always look into shared ownership. Naturally, they'll have to pay rent for the half they don't own, but if they complain about that your DP just needs to tell them that they've had an amazingly generous free ride so far, but exactly how long did they expect to have somebody else paying half of their mortgage with no financial benefit to themselves? They can hardly plead poverty if they're already living in a very decent family home and they want to move to something even bigger. If anything, they should be cutting their cloth and moving to somewhere smaller.