It is indeed a fact, based on the documents released by the FOIA, that both FBI and OSS (the precursor to CIA - CIA didn't exist at that time) obtained papers from Nikola Tesla after his death in 1943.
Some of the released material show that the Department of Defense (DoD), among others, was interested in these papers because:
"SUBJECT: Papers recovered on the Death of Nicola Tesla (U).
(U) We understand that FBI may have possession of a number of papers
found after the death of Nicola Tesla in 1943. [...] (C) We believe
that certain of Tesla's papers may contain basic principles which
could be of considerable value to certain ongoing research within the
- letter written to FBI by LtCol. Allen J. MacLaren, 8. Feb 1981
Another document dated 12 Jan 1943 shows that a vast amount of material including papers was seized from the hotel:
"[...] On Saturday afternoon, Jan nine, Corsuch and -- Fitzgerald of
Alien Property Control went to hotel and seized all the property of
Tesla, consisting of about two truckloads of material, sealed all
articles and transferred them to the Manhattan storage and warehouse
Co. NY, where they are now located. [...]"
- Page 1 (scan), Page 2 (scan)
Another letter to FBI dated April 20, 1976, shows:
"[...] After Tesla's death, scientists from the Navy and OSS performed
a cursory examination of the diary and notes [...]"
- Page 1 (scan)
There exist other released documents as well, not included here.
I don't think there is doubt about the confiscation itself. The speculations are more about what information the papers contained as well as what items where found.
It's also a fact that J.P. Morgan wanted to finance Tesla (see Unsuccessful ventures / Energy) as he was researching if he could utilize the ionosphere - where the idea of "free energy" comes from. If the experiments were successful or not is unknown, but eventually Morgan pulled his financial backing.
Further, the speculations refer to the High Frequency Active Auroral Research Program (or "HAARP" which you find in Alaska) as they also utilize the ionosphere - this is the basis of suspicion of HAARP because their research is based on Tesla's work. Considering the letter above from the DoD/USAF this is perhaps not so out of the blue (pun intended). From the official HAARP page they answer:
"[...] Because the DoD operates numerous communication and navigation
systems whose signals either depend on reflection from the ionosphere
or must pass through the ionosphere to satellites, there is obvious
DoD interest in understanding the ionosphere's effect on these systems
to improve their reliability and performance. [...]"
- FAQ, HAARP official page (see last question)
The part that brings in the "conspiracy" element is what HAARP is being used for, and there is a question about it on this site. This however is not related to Tesla.
One thing is for sure, Tesla was a great genius almost "forgotten" in history.
UPDATE 10/2016 FBI recently released (by FOIA) a batch of related documents that can be read here: