Given that these two periods are over a century apart, the simplest answer to your question is that there is little direct connection between them or immediate influence of the one upon the other. The Restoration was the era beginning with the reestablishment of the monarchy with Charles II in 1660. The Romantic period, though certain writers had earlier anticipated its trends, is usually considered as beginning in the late 1790s, in Britain with the publication of Wordsworth and Coleridge's Lyrical Ballads.
That said, as with everything else in the history of the arts, the past, no matter how distant, can always be seen to have some degree of connection with any movement or tendency. The greatest poet of the Restoration was John Dryden. In two respects, Dryden was a direct and an indirect influence or inspiration upon at least one of the most important Romantic poets, Lord Byron. Dryden was an exponent of a type of political and literary satire that set the standard for the poets who followed him: Alexander Pope in the eighteenth century, and Byron in the early nineteenth. The tone and even the wording of Dryden's "Mac Flecknoe" is echoed in Byron's early satire "English Bards and Scotch Reviewers," and the allegorical political satire of Dryden's "Absalom and Achitophel" and "The Hind and the Panther" has an analogue nearly a century and a half later in Byron's "The Vision of Judgment." One can conclude, more generally, that the free-wheeling and explicit tone Dryden and other poets of his time adopted was what ultimately made possible the freedom of expression characterizing all English literature from that point on, including that of the Romantic period.
The second major point about the Restoration is that it was during this period that the English first developed a historical sense of their own international greatness as a literary nation. Dryden regarded John Milton as not only the greatest English poet, but as the one who superseded the epic poets of antiquity, when he wrote:
Three poets, in three distant ages born,
Greece, Italy and England did adorn.
The first in loftiness of thought surpassed,
The next in grandeur, and in both, the last.
The force of nature could no farther go:
To make a third, she joined the former two.
Dryden is here saying that Milton, in Paradise Lost, surpassed the poetic greatness of both Homer and Virgil. This is a notion fully embraced especially by the English Romantic and Victorian writers. It was first during the Restoration that the sense of England as having a literature that could compete on the world stage with other nations began.
The Restoration is marked by the restoration of the monarchy to the throne of England. This period is not generally connected by commentators to Romanticism. The Restoration is usually cited as a period from 1660-1700. Romanticism, at its earliest, begins with writers like Blake near the end of the 18th century, and isn't a literary force until the publication of Wordsworth's and Coleridge's poetry collection, Lyrical Ballads, until 1798.
The period between the Restoration and Romanticism caused the romantic revolution, not the Restoration itself.
In short, there isn't an often spoken of and written about cause and effect relationship between the two periods you ask about.
The Restoration did result in the reopening of theatres, and led to innovations such as women being allowed on stage. It also brought about freedom of performance, in that plays became bawdy and somewhat sexually oriented.
But beyond that, I'm unaware of any direct connection. Maybe another editor will know more than I do.