Is there a term for an externality in which prices for a good go up due to consumers being subsidized to purchase or consume that good?
That's a standard case of a Consumer Price subsidy. Its effect will be that of market expansion: higher quantity at higher equilibrium price, irrespective of whether it will be lump-sum (a fixed monetary amount) or a fixed proportion of the price.  

A) LUMP SUM SUBSIDY
Consider a monopolist. The monopoly faces a downward-sloping demand curve, say

$$q_d = a -b(p-S) = (a+bS) - bp$$

where $S$ is the lump-sum subsidy. Here the demand schedule shifts upwards by $bS$ maintaining the same slope. Profits are (assume linear costs for simplicity)

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [(a+bS) - bp]$$

and the first order condition  for a maximum gives

$$(a+bS) - bp -b(p-c) = 0 \implies p^* = \frac {a+ bS +bc}{2b} = p^*|_{S=0} + \frac {1}{2}S$$

Profit-maximizing price is increasing in the lump-sum subsidy. Now substitute this into the demand equation

$$q_d^* = (a+bS) - b\frac {a+ bS +bc}{2b} = \frac {a-bc}{2} + \frac{b}{2}S = q_d^*|_{s=0} + \frac {b}{2}S$$

Quantity demanded also increased. So the lump-sum subsidy leads to market expansion.  

B) PROPORTIONAL SUBSIDY
Here we have 
$$q_d = a -b(1-s)p,\;\;\; 0\leq s&lt;1$$
The demand schedule changes slope (becomes more "flat"), maintaining the same intercept. 

$$\pi = p\cdot q_d -c\cdot q_d = (p-c)\cdot [a -b(1-s)p]$$

and the first order condition  for a maximum gives

$$a -b(1-s)p -b(1-s)(p-c) = 0 \implies p^* = \frac {a+ b(1-s)c}{2b(1-s)} = \frac {a}{2b(1-s)} + (c/2)$$

Equilibrium price is (non-linearly) increasing in the proportional subsidy. Substitute this into the demand schedule to get

$$q_d^* = a -b(1-s)\frac {a+ b(1-s)c}{2b(1-s)} = a -\frac {a+ b(1-s)c}{2} = \frac {a-bc}{2} + \frac {bc}{2}s$$

Here too equilibrium quantity is increasing in the subsidy.  

For a sector like Health services, where increased market access is of concern (in some countries) due to societal values (and/or cold calculations as regards the material benefits to society in having more healthy members), and where "quantity demanded" is not based so much on desires but rather on not-negotiable needs (illnesses, accidents, etc), the rationale for such subsidies is that the quantity increased will mainly come from new "consumers", previously excluded from the market due to the pre-subsidy equilibrium price in combination with their income level.