Assuming the information in Debrett's is up to date, the line of succession to the title after Lord Bingham is:
1. Hon Hugh BINGHAM (b 1939), brother of the 7th earl. Unm.
(Descendants of the 5th Earl:)
2. Nicholas Charles BINGHAM (b 1943), m with issue
3. Charles Nigel BINGHAM (b 1979), m 2011, no issue (son of No. 2)
4. Peter John BINGHAM (b 1945), brother of No. 2, m with issue
5. Philip Charles BINGHAM (b 1974), m 2001, no issue (son of No. 4)
6. Richard Patrick BINGHAM (b 1978), m 2011, no issue (son of No.4)
7. David Julian BINGHAM (b 1951), unm, (brother of Nos 2 & 4)
Nos 3 & 6 have only recently married (2011) and may well produce further heirs for the title.
Even supposing Lord Bingham dies without producing heirs, we may well expect one of his younger cousins to survive him and succeed as earl, in which case the title won't become extinct until the death of the final male heir to the title. (Note: it will only become DORMANT if there is a strong belief that there are untraced heirs to the title. There's no suggestion that is the case with regard to this earldom.)