Not necessarily. In the sense of "last" being synonymous with "remaining" rather than "final", the last Jedi is Luke, or possibly the last Jedi are Luke and at least one surviving disciple of his who is also a Jedi Knight, the disciple being a returning character like Rey or a previously hidden new character. While Rey wasn't known to be a Jedi in The Force Awakens, she could possibly become a Jedi Knight by the end of The Last Jedi, just as Luke did by the end of Return of the Jedi. Kylo Ren could also return to being Ben Solo, who might've been a Jedi Knight before turning to the dark side of the Force. Furthermore, the idea of Luke being the last Jedi isn't new. In Return of the Jedi, Yoda tells him, "When gone am I, last of the Jedi will you be." In The Force Awakens, there are two references to Luke being the last Jedi.
As for the prospects of the Jedi going extinct, it's hardly outside the realm of possibility, but it's also hardly a certainty. (The film's first trailer reveals an intention indeed, and on the part of a Jedi, to end the Jedi.) If users of the Force go extinct altogether (whether Jedi, Sith, "Gray Jedi", "Dark Jedi" or the Sith-esque), the ramification is that the people and beings of the galaxy will have nothing more than a faintly practical and largely superstitious connection to the Force, as in no knowledge of how to use it to perceive or manipulate the environment like the Jedi and Sith had. The users of the Force could become subjects of mythology, whilst things like the dark cave (on Dagobah, in The Empire Strikes Back) would be subjects of investigations into miracles and paranormal events. To some extent, such had already happened sometime before the events of the Star Wars original trilogy. Even throughout time era of the prequel trilogy storylines, the public awareness of scope of Jedi powers was minimal, given the arcane nature of monastic orders as well as the Jedi's reluctance to be showoffs.