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So the most obvious evidence of the "Time loop" in the Evangelion series is the difference between the Original series and the Rebuild series. I.e. in both the opening sequence is the same, but the major difference is that the ocean is red in the Rebuild series, following the events of the Third Impact. The rest of the series leads to an alternative ending.

A bit of a crackpot theory is that all iterations of the Evangelion series (including games, dating apps, manga, etc.) are all alternate iterations of the same time sequence, just panning out differently and leading to alternate timelines and events.

Is there any evidence this is the case, and if so, what causes the loop?

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There is a very detailed entry relating to all the various forms of Evangelion on the SF-Encyclopedia.com website, which while not exactly offering the canon confirmation you are seeking for your theory certainly discusses it very seriously.

You can read the whole entry here... http://www.sf-encyclopedia.com/entry/shinseiki_evangelion

but these quotes do seem to confirm, or at least agree with your theory that all iterations of Evangelion can be considered variations of a time loop.

Ten years later, the story was retold as a four-part animated film series, Rebuild of Evangelion, beginning with Evangelion 1.0: You Are (Not) Alone (2007; original title Evangerion Shin Gekijōban: Jo). Despite the outward appearance of a simple summary of previous episodes with some bonus footage, the first film offered rich pickings for fans who remembered the original, presenting radically different assessments of many lead characters. Its sequel, Evangelion 2.0: You Can (Not) Advance (2009; original title Evangerion Shin Gekijōban: Ha) departed more radically from the original storyline, but contained background details that appeared to recall it, suggesting that the films were not a remake at all, but a sequel, set aeons after the original (see Time Abyss).

also from the same article

Tanslation idiosyncrasies, still in contention 25 years later in the serial's Netflix release, and the chaos of the show's original denouement, both onscreen and behind the scenes, have largely obscured its relatively simple plot, which was only truly confirmed and clarified in the materials accompanying the PlayStation 2 Videogame Shinseiki Evangelion 2 (2003). The Forerunners Adam and Lilith (see also Adam and Eve) are revealed as rival biological agents of Panspermia, arriving on Earth in the distant past. With Adam placed in stasis (see Stasis Field), as two planetary seeders cannot co-exist on the same world, Lilith proceeds to populate the Earth with her own genetic material. The Second Impact that precedes the television series is revealed, in the briefest of asides, as the accidental awakening of Adam by human Scientists, causing a global cataclysm and the subsequent onrush of Adam's "children", the so-called Angels, held off with man-made bio-Weapons deriving in part from Lilith's DNA. The seventeenth Angel, Kaworu, created from Adam's own genetic material, realizes that it is Lilith and not Adam who has Terraformed the planet, and voluntarily sacrifices himself, on the grounds that humanity (as represented by Shinji) is essentially worth saving. Much of this backstory, however, is only dimly apprehended in the television series, often as mere snatches of dialogue, since humanity is more concerned with its own Ecological and societal problems, Disaster relief, and a subplot in which a secret organization (see Secret Masters) attempts to effect the Uplift of humanity. In a quintessence of Little, Big contrast [see The Encyclopedia of Fantasy under links below], the final two episodes of the original series withdraw from the apocalypse into the mind of Shinji as he frets over the nature of becoming one with the Posthuman Hive Mind. The story concludes with him on the cusp of his decision, thereby creating a confusion of divergent timelines, some of which investigate the possible results of such a Jonbar Point.

Note the use of the term "Jonbar point" which seems to tally very closely with what you are suggesting.

The SF- Enclyclopedia.com cannot exactly be considered canon as by nature an encyclopaedia does not contribute to the content of a series/franchise or validate theories about any of its subjects. It simply presents known facts on a subject and cross connects points where possible. That said its editors, John Clute, David Langford and David Nicholls should certainly be considered authorities on SF and their viewpoint noteworthy.

The above referenced article mentions Hideaki Anno's disillusionment with fandom and his unwillingness to engage with it these days, which makes it unlikely that there will be any direct quote from him on this subject. At the time of writing this I certainly have not found one but will update my answer if I do.

To sum up, yes, it would seem your assumption that all iterations of Evangelion are different variations of the time loop. But due to the (possibly deliberate) elliptical approach to the narrative we might never know the details of the cause.

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