There are a number of Unicode characters categorized as "Extended Pictographic" which have no color and usually appear smaller than "normal" emojis. Here are some examples:
☺ ☻ ♥ ♦ ♣ ♠ ♂ ♀ ♪ ♫ ☼ ↕ ↔
These don't have a full-colour emoji counterpart. Does the Unicode Consortium provide any table or other information that allows me to identify these characters, i.e. to distinguish these monochrome "Extended Pictographic" characters from full-colour "Extended Pictographic" characters? I wasn't able to find such information myself.
The distinction you're seeing is called "emoji presentation" vs "text presentation." Some characters have both, and may have one or the other be the default.
The file you want is emoji-data. When you say "these don't have a full-colour emoji counterpart," that's not correct for most of them. They just have default text representation. I'll walk through a few of these to see how to understand them. One of my favorite exploration tools for this is the Unicode Utilities. You'll want UTS#51 as well.
(Note that in Unicode, WHITE means "not filled in" and BLACK means "filled in" in many cases for historical reasons, going back to Japanese flip phones. They are not actually colors. Similarly, HEAVY means "bold" or "wide.")
So there is an emoji form of this, but it's not the default. We have to ask for it by adding U+FE0F (VARIATION SELECTOR-16). With that, this character displays as ☺️.
For comparison, see this character, which is a more "traditional" emoji.
So, this is not an emoji at all which means it has no "emoji presentation." It is merely "similar in kind to characters with the Emoji property" (i.e. extended pictographic).
Another example of an Emoji without Emoji_Presentation. The emoji form is ↕️.
And just for a little completeness:
Digits are also emoji, and can take VS-16 modifier:
If you want a browsable list of characters with different properties, see the Character Property Index.