This is not really about the word to or the verb meet. Rather, the phrase is unlikely combined with a to-infinitive (in this case, "to meet") tells us that the action of the infinitive verb will probably not happen in the future.
The words likely and unlikely are about how certain we are about something happening in the future. From Cambridge Dictionary's English Grammar Today1:
Likely and unlikely are adjectives. We use them to say that something will probably happen or not happen
in the future. We can use them before a noun, or with the verbs be, seem and appear . . . . We can follow likely or unlikely by a verb in the
The economy is likely to recover slowly after the long recession.
Are you likely to want this cardboard box or shall I throw it out?
The weather seems unlikely to change over the next few days.
As noted in this explanation, if we say an event is [un]likely we are talking about how certain we are that the event will happen in the future. So in your example,
[The government] is unlikely to meet the full cost.
the sentence is talking about how confident we are now that the government will pay for the whole thing at some time in the future.
Note that if we used a different tense of the verb with unlikely, we could have a different meaning:
[The government] seemed unlikely to meet the full cost.
In this case, we've paired unlikely with the past tense of the verb "seem", so the sentence means that sometime in the past it looked like the government would be unable to pay. In that case, the expected failure to pay was at some time in the future of the time of "seeming" but not necessarily later than now. To make this clearer we can add a bit of context:
Ten years ago, the government seemed unlikely to meet the full cost. However, there was very strong public support for the project, so a few years later the government promised to fund the entire project, and by last year it was completely paid for.
Similarly, we can use different infinitive verbs with these same likely and unlikely structures:
The government is unlikely to surrender.
The government seems likely to raise taxes soon.
Last year, the government appeared likely to lower interest rates.
You can read more about the use of likely and unlikely in the article linked above.
As was pointed out in comments, unlikely can be paired with other forms of infinitive besides the simple infinitive (e.g. the continuous—is unlikely to be meeting—and so forth) which of course changes the meaning. As usual, the infinitive in this case takes its tense from the verb that precedes it, and contributes aspect—the relationship in time between the action of the preceding verb and the infinitive verb.
The key issue here is that the usual temporal relationship for simple to-infinitives is changed when we add likely or unlikely. With the simple infinitive, ordinarily the action of the infinitive and the action of the tensed verb are understood to happen at the same time. With (un)likely, however, the infinitive's action is shifted to happen after the tensed verb. Compare:
They seem happy to see me (Seeming happy and seeing me are both happening in the present); and
They seemed happy to see me (Seeming happy and seeing me both happened in the past)
They seem likely to see me (Seeming likely is happening in the present, but seeing me hasn't happened yet—it probably will happen in the future, but might not); and
They seemed likely to see me (Seeming likely happened in the past, and seeing me was then expected to happen at some time later than that—possibly a less-distant past, possibly the present, possibly still in the future).
With other forms of infinitive the relationship is more complicated, and outside of the scope of this Q&A, but generally the "action" that is shifted forward is finding out the truth or falsity of the infinitive.
I am happy to have met him. (Happiness is in the present; meeting him is in the past.)
I am likely to have met him. (Likeliness is in the present; meeting him or not is in the past; and finding out whether meeting him happened is in the future.)
You can learn more about the different aspects of infinitives in the Grammaring article linked above, or your favorite English grammar source.
1 Markdown formatting added; sorry it's so ugly, but I couldn't think of a better way of adding emphasis when the original is already using italics and bolding.