Hi, William. Count me in with Deb on this one: the origin and etymology of the Tease surname are very probably not French. I believe the consensus would be that it's of place-name origin taken from the River Tees in northern England.
One of the surname DNA projects I administer is a variant spelling, Teasley, with relatively significant Colonial American roots and the subject of a 1990 book by Kyle Williams (no relation...that we know of). There are loads of variant spellings, both with and without the -ley (an old Anglo-Saxon suffix, which meant a homestead or meadow).
The River Tees has its source in northern England on the east slope of Cross Fell, about 12 miles east-northeast of Penrith, and it meanders its way east for over 85 miles to the coast of the North Sea at Teesmouth National Nature Reserve, not far from Middlesbrough...about 20 miles southeast of Durham...which I just had to thrown in here for Deb's sake. Before the 20th-century reorganizations of some of England's historic counties, the river was the demarcation between Durhamshire and Yorkshire.
David Simpson says that the oldest place-names in the region are those of the rivers, and with Tees he includes Aln, Don, Derwent, Deerness, and Tyne as examples. He speculates that the names are at least of pre-Roman Celtic origin, possibly even thousands of years ago when the lingua franca of the area was even earlier than the bifurcation of Indo-European languages into Germanic > West Germanic > Anglo-Frisian > Old English and Celtic > Insular Celtic > Goidelic and Brittonic.
Simpson says: "The River Tees is thought to have a Celtic river-name though its roots may be earlier. It's related to an Old Welsh [classified linguistically as a Brittonic language; sometimes referred to as P-Celtic] word for 'heat' and means 'boiling, surging river' perhaps alluding to the waterfalls of upper Teesdale like High Force."
In a 2016 article for the BBC, James Harbeck writes: "British history didn't start with the Celtic peoples (Stonehenge didn't build itself, after all). But the Celtic tribes that arrived during the Iron Age, which started around 800 BC, were the first to give a clear linguistic contribution that has lasted to modern times. They came in groups from the continent; those in the north spoke Goidelic (the source of Gaelic), while southerners spoke Brittonic.
"Even today, many hills and rivers have kept their Celtic names -- especially in the north and west. The Wrekin takes its name from Celtic. So do about two-thirds of England's rivers: Avon, Derwent, Severn, Tees, Trent, Tyne... Some of these names may even have come from the people who were here before the Celts."
Another interpretation is that the word "tee" is derived from the Scottish Gaelic word "taigh" meaning house. This is often shown as the derivation of the term "tee" when used pertaining to the golf implement or the platform used to hold the ball in American Football or Rugby. I think this can be safely discounted regarding the name of the river, though; "taigh" would be far too new as an etymology. Most historians and linguists believe that Gaelic was brought to Scotland by the Dál Riata from Ireland who settled on Scotland's west coast circa 400-500 AD, approximate to departure of the Romans. Gaelic in Scotland stayed mostly confined to western coast until around the 8th century, when it began expanding north and east. The River Tees had borne that name for many centuries before the expansion of Scottish Gaelic.
John Grenham's website shows several Tease (and variant spellings) families in Ireland at the start of the 20th century, but it seems probable the surname was first used in England. Most of the existing census records in Ireland prior to 1901 were destroyed in the 1922 fire, so unfortunately we have very little to go on before that as to surname frequency. In the 1901 census, there were 90 heads-of-household enumerated with the last name "Tease"; 80 of the 90 were in County Donegal.
Given the history and timing of the Huguenots in Ireland, it would seem improbable that the Tease/Tees surname came from France; in fact, some quick Google-fu turns up no French names with those spellings. If there is an Irish and Huguenot tie-in, I suspect it would have been a Tease family emigrating from England to Ireland, perhaps during the Reformation and so-called counter-reformation shaking up more than just organized religion from roughly the time of the inauguration of Elizabeth I in November 1558 through the relatively short-lived reign of the Dutch Protestant William III of Orange. The Irish Parliament, after the restoration of Charles II, passed a law in 1662 to encourage French Protestants to come to Ireland. When James II was booted out in the "Glorious Revolution" of 1688, the Irish parliament rescinded the Oath of Supremacy to the Church of Ireland, and that opened the door for a new influx of French Protestant refugees.
Some French Huguenots thrived in Ireland, both in political and economic influence, and some chose the more religiously permissive shores of the New World. But my guess is the Tease origin is from northern England; that any connection to French Huguenots in Ireland was by marriage or association, not by emigration from France. Just a guess, though.