What is the sum of Fibonacci reciprocals?

This is A079586, where you can find several references. It doesn't look like there is a 'nice' closed form, but some results have been proved. The constant is irrational [1] and can be computed rapidly [2], [3] with various methods.

[1] Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.

[2] Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

[3] William Gosper, Acceleration of Series, Artificial Intelligence Memo #304 (1974).