Probably, but it depends on your DM.
More specifically, the dart certainly is an object, as specified in What is considered an object?.
Concerning the size of the object, we have some precedence for object sizes already.
The DMG has a table on object hit points on page 247 (the table can also be found on the official DM screen). In this table, "bottle" and "lock" are listed as examples of tiny objects. Examples for small objects, the size category above "tiny", are a "chest" or a "lute".
Furthermore, in the PHB in the equipment section, Ball Bearings are specifically described as "tiny metal balls", and a Holy Symbol can be a "tiny box holding [...]".
In addition, a variety of spell components are described as "tiny".
Overall, we still don't have a clear definition of object sizes as we do for creature sizes. It's unclear if the creature size rules apply to objects as well, since creature sizes don't refer to their actual size, but to the "space a creature of a particular size controls in combat" (PHB, p. 191).
If object sizes are equivalent to creature sizes, tiny objects would be up to 2.5 x 2.5 ft. big (PHB, p. 191).
If not, you can refer to the examples listed above to determinate which kinds of objects would be considered tiny.
Overall, however, a dart probably counts as tiny either way, so you should be able to use it for the Artificer's Magical Tinkering feature. If you want to be safe, talk to your DM first, since he or she is the final arbiter on the matter.