I think you may have discovered a bug in TubePress (lucky you!). Here's what's happening:
When you click a thumbnail (or the Ajax-enabled pagination buttons) TubePress makes an asynchronous request to fetch the new content. Here's an example URL that it might try to fetch after you click a thumbnail:
http://atat.ro/wp-content/plugins/tubepress/sys/scripts/ajax/playerHtml.php?tubepress_video=_N26Ztax25A&tubepress_shortcode=%255Btubepress%2520resultsPerPage%253D%252228%2522%252C%2520filter_racy%253D%2522strict%2522%252C%2520output%253D%2522searchResults%2522%252C%2520searchProvider%253D%2522youtube%252Bvimeo%2522%252C%2520ajaxPagination%253D%25221%2522%252C%2520cacheEnabled%253D%25221%2522%252C%2520likes%253D%2522%2522%252C%2520tags%253D%25221%2522%252C%2520rating%253D%2522%2522%252C%2520mode%253D%2522tag%2522%252C%2520tagValue%253D%2522Romania%2520%252B%2520stiri%2520%252B%2520romani%2522%252C%2520playerLocation%253D%2522popup%2522%252C%2520length%253D%2522%2522%252C%2520paginationAbove%253D%2522%2522%252C%2520views%253D%2522%2522%252C%2520description%253D%25221%2522%252C%2520descriptionLimit%253D%2522100%2522%252C%2520paginationBelow%253D%25221%2522%252C%2520playerImplementation%253D%2522longtail%2522%252C%2520thumbWidth%253D%2522120%2522%252C%2520thumbHeight%253D%252290%2522%252C%2520orderBy%253D%2522newest%2522%252C%2520fullscreen%253D%25221%2522%252C%2520theme%253D%2522myposts%2522%252C%2520galleryId%253D%2522688014259%2522%252C%2520sequence%253D%2522%25270MvmT3HALWg%2527%252C%2520%2527_N26Ztax25A%2527%252C%2520%25273K3kaszbEjE%2527%252C%2520%2527HNUTyC-mDlI%2527%252C%2520%252730ErnKwRNXc%2527%252C%2520%2527XY2Vu36MXsg%2527%252C%2520%2527SqZ9-7_ooeo%2527%252C%2520%25279LF_CeRz2YA%2527%252C%2520%2527NyQFkH-nstg%2527%252C%2520%2527_f-xxmQp2Ww%2527%252C%2520%2527SQt5NjHG12M%2527%252C%2520%2527P-kvTbl4VDg%2527%252C%2520%2527ljR5ZJp4yt8%2527%252C%2520%2527pt9NXtwThDA%2527%252C%2520%2527exR1vDanU5M%2527%252C%2520%2527B1vWJqJzZnc%2527%252C%2520%2527TmeBPGCkH3k%2527%252C%2520%2527kwnNW-5e_YY%2527%252C%2520%2527ZqShrAcDlDc%2527%252C%2520%2527cwQFw8as4cQ%2527%252C%2520%2527c8dPvxZqqe4%2527%252C%2520%2527Tg1R1cUoA-k%2527%252C%2520%2527yrp3nvQ1u6g%2527%252C%2520%2527o8cnEKWDr2U%2527%252C%2520%2527SaC_okewgNA%2527%252C%2520%2527rHMu7WAEBx8%2527%252C%2520%2527E1p1Ftpf1DI%2527%252C%2520%25272msl2hb-SYQ%2527%252C%2520%2527h42pQ7BVmh8%2527%252C%2520%25273ImW5foC7jU%2527%252C%2520%2527_h1KSBK4suA%2527%252C%2520%2527ry-DAFusCj8%2527%252C%2520%2527qPKwyPDrd6U%2527%252C%2520%2527VyaV9vxh1_0%2527%252C%2520%2527dsKfTDCHmKA%2527%252C%2520%2527XigAfN6jVE8%2527%252C%2520%2527V4_IThaKqOE%2527%252C%2520%2527d-D-YXVYFyM%2527%252C%2520%2527ONvQB6bZCi0%2527%252C%2520%2527ij_ZyegT6XQ%2527%252C%2520%2527QLzwgbRj1h4%2527%252C%2520%2527VBdFukgMlS4%2527%252C%2520%2527dv6HabwD_vE%2527%252C%2520%2527rl9us98DN8M%2527%252C%2520%2527hMUWYMFfNV8%2527%252C%2520%2527P0LoRh6LLtk%2527%252C%2520%2527WZbFguQSkwo%2527%252C%2520%2527YgStQKixuos%2527%252C%2520%2527mEMFmzAfVZg%2527%252C%2520%2527VTbc8TxJqu0%2527%252C%2520%2527PTf3tBv3svg%2527%252C%2520%2527QqWBiarXuW4%2527%252C%2520%2527aISpe9nhUTU%2527%252C%2520%2527hS7TIlxxkkQ%2527%252C%2520%2527sevBzHh7Pcs%2527%252C%2520%2527SJeY8N_6Yqk%2527%252C%2520%2527fTUzT6wqy4o%2527%252C%2520%2527fPIZBaxShXI%2527%252C%2520%2527r6Dc61j8HCk%2527%252C%2520%2527gpcGLv58lek%2527%252C%2520%2527hwcs_4442BM%2527%252C%2520%2527h1xTPAJeerw%2527%252C%2520%2527fwzIz0dc08E%2527%252C%2520%2527uFMgFdZqADg%2527%252C%2520%2527MrWgg-Czlxc%2527%252C%2520%2527p3TtiJDCOkE%2527%252C%2520%2527LMwMZh-Ow6s%2527%252C%2520%2527ZvVJkjnsbro%2527%252C%2520%2527TYZBLnSYW-c%2527%252C%2520%2527EDFGGpBM2uQ%2527%252C%2520%2527U5dGIBi8Zgk%2527%252C%2520%2527PHLr3EdNstw%2527%252C%2520%2527NxmVUVKghQw%2527%252C%2520%25273dUUCHZCzVA%2527%252C%2520%2527M_XfSDRrJB8%2527%252C%2520%2527SF4Y8CRkub8%2527%252C%2520%2527yvZEAajlrQU%2527%252C%2520%2527hzntsBfSq90%2527%252C%2520%2527jHjCCcKYOVE%2527%252C%2520%25279d23DYnqWGI%2527%252C%2520%2527AaDbVqbCHvY%2527%252C%2520%2527e99MJo886gY%2527%252C%2520%2527XQ6U1DEpric%2527%252C%2520%2527z6CkkEAu-uA%2527%2522%255D
All that extra stuff appended to the URL (after the ? character) is the TubePress shortcode in use. Normally, this is all fine. But in your case this request results in an HTTP 402 error, probably because the length of the URL with its associated parameters is too long. To prove this, notice that if we grab the same URL without all the extra query parameters:
http://atat.ro/wp-content/plugins/tubepress/sys/scripts/ajax/playerHtml.php?tubepress_video=_N26Ztax25A
then we get correct output.
The solution is really for TubePress to start using HTTP POST instead of GET when making Ajax requests. I've
filed a bug ticket that you can star. This will take some time to implement, but in the meantime here's a solution that
should work for the time being.
That should dramatically reduce the length of the URL used. Could you give it a try and let us know?