Assuming you don't count connection set-up (as you indicated in your update), it strongly depends on the cipher chosen. Network overhead (in terms of bandwidth) will be negligible. CPU overhead will be dominated by cryptography. On my mobile Core i5, I can encrypt around 250 MB per second with RC4 on a single core. (RC4 is what you should choose for maximum performance.) AES is slower, providing "only" around 50 MB/s. So, if you choose correct ciphers, you won't manage to keep a single current core busy with the crypto overhead even if you have a fully utilized 1 Gbit line. [Edit: RC4 should not be used because it is no longer secure. However, AES hardware support is now present in many CPUs, which makes AES encryption really fast on such platforms.]
Connection establishment, however, is different. Depending on the implementation (e.g. support for TLS false start), it will add round-trips, which can cause noticable delays. Additionally, expensive crypto takes place on the first connection establishment (above-mentioned CPU could only accept 14 connections per core per second if you foolishly used 4096-bit keys and 100 if you use 2048-bit keys). On subsequent connections, previous sessions are often reused, avoiding the expensive crypto.
So, to summarize:
Transfer on established connection:
First connection establishment:
Subsequent connection establishments: