IIRC, the Lavabit case was one of the reasons (and perhaps the strongest one) for the current push towards forward secrecy cipher suites in TLS (with TLS 1.3 even having them as the only option).
Since Lavabit used the older RSA key exchange, instead of a DHE key exchange, the private key they were forced to hand over could decrypt all past HTTPS connections. With forward secret (DHE/ECDHE) key exchanges, the private key can only be used to impersonate the server; it cannot be used to decrypt any past (or future) HTTPS connections.
Had Lavabit used DHE/ECDHE, they would have nothing to give. Even adding a backdoor to their servers (or a MITM logging middlebox) would gain nothing, unless the target logged in after it was installed. They would be able to simply respond "we have nothing useful to give" and be done with it, like recently happened to Whispersystems.
Since Lavabit used the older RSA key exchange, instead of a DHE key exchange, the private key they were forced to hand over could decrypt all past HTTPS connections. With forward secret (DHE/ECDHE) key exchanges, the private key can only be used to impersonate the server; it cannot be used to decrypt any past (or future) HTTPS connections.
Had Lavabit used DHE/ECDHE, they would have nothing to give. Even adding a backdoor to their servers (or a MITM logging middlebox) would gain nothing, unless the target logged in after it was installed. They would be able to simply respond "we have nothing useful to give" and be done with it, like recently happened to Whispersystems.