The Magic Initiate feat description reads, in part (emphasis mine):
You learn that spell
and can cast it at its lowest level. Once you cast it, you must finish
a long rest before you can cast it again using this feat.
There is typically no way to upcast a 1st-level spell that is gained via the Magic Initiate feat (see below for exception).
A 1st-level spell gained via the Magic Initiate feat does not typically use a spell slot (again, see below for exception). This makes sense, as classes without spell slots can take this feat.
Your player should keep this inability to upcast in mind when selecting his 1st-level Magic Initiate spell. If he is planning on advancing his character to high levels, there are better choices than a basic damage-dealing spell like burning hands.
Alternately, if he really wants burning hands and wants to be able to upcast it, he could always take a level in a class that has that spell on its spell list (assuming you allow multiclassing, and the multiclassing requirements are met).
In his answer, J.A. Streich points out that the Sage Advice Compendium (PDF link) states that if you take the Magic Initiate feat choosing your own class, then you can use spell slots to cast the 1st-level spell you learned with Magic Initiate. So if your player took Magic Initiate (cleric), then he would be able to use spell slots to cast whatever 1st-level cleric spell he chose.
The Sage Advice Compendium ruling doesn't discuss multiclassing, but I don't see why it wouldn't apply to multiclassing too: If your cleric player took Magic Initiate (sorcerer) and burning hands, and later took a level in sorcerer through multiclassing, he should then be able to use spell slots to cast burning hands.
This seems like suboptimal feat selection to me, but it is an option.