The reason my last bass...the one I'm unloading...had an extended B was because I wanted the E and A strings non-speaking length to be the same as a 4 string bass in an effort to "slop" up the E. That left the last position as the only option. We were just using characteristics of 4 string basses as much as we could...whatever we could.
There is no difference, given the same size string, in static tension (tension while the string is sitting still) between the extended B and a standard B. There's no way there can be. Physics doesn't allow for it.
That doesn't speak to the dynamic effects of the straighter and longer non-speaking length, nor does it address the relationship between transverse mode of vibration (a function of string mass/stiffness, length, and tension) and longitudinal vibration (a function of only string mass/stiffness and length). The termination points of the string stop only the transverse vibration. They don't (particularly the nut) stop the longitudinal mode of vibration, and the relationship between those two modes is a part of "how the string sounds"...it lowers the pitch of the longitudinal mode because of more length). Tuning your E down to D is a good way to demonstrate this. The string "sounds different" in large part because the longitudinal mode has not changed due to lower tension, but the transverse mode (the one we hear as the fundamental) has gone down a whole step. I don't think anyone would argue with that example. What we hear as "overtones" is often the longitudinal mode, which is significantly higher (a few octaves, often) than the fundamental.
Another note is, traditionally, longer scales on string instruments were so that the string could have a lower mass per unit length, and still be the SAME tension as a shorter one. This produces a clearer fundamental and less "thud" or inharmonicity (components of the sound that are not related to the pitch of the fundamental through the overtone series)
So basically, it doesn't make the string tension higher. That's irrefutable. But it may, in fact, have a tonal effect, and may offer a "sense" of greater tension due to the extra length beyond the nut stretching as the string is pulled, released, and vibrated. There is NO evidence anywhere that I've ever been presented with that suggest that it does NOT do these things, and PLENTY that supports it.