Does it matter which or how many monster parts I use in an elixir?

The effect of the elixer (e.g. Fireproof, Attack Buff, Defence Buff) is governed by the number of ingredients used of that type. 1 of an ingredient will give level 1, 2 or 3 of an ingredient will give level 2, and 4 of an ingredient will give level 3.

As far as I have seen, 4 ingredients of the same type are required to make a Tier 3 Elixer. The actual items used don't seem to matter, just that they all buff the same stat.

The monster item governs how long the effect lasts. Low tier items, such as Bokoblin Horns, will only last a few minutes, where as high tier items, such as Star Pieces and the Fangs of the Dragons, can last up to 30 minutes.

This gallery has a pretty great breakdown of all the effects of all monster parts (and everything else) - generally speaking, most of them add duration, with shards of the dragon horns doing the best work at a whopping thirty minutes.

Adding several ingredients that have additional effects at higher "levels" will give a better bonus to food/elixirs. For example, three mighty bananas, a mighty thistle, and a shard of one of the dragon's horns will provide a +3 damage boost for thirty minutes.

Monster parts themselves do not appear to have much of an effect aside from duration increases (some dragon parts adding hearts notwithstanding).