Turn a blind eye
To knowingly refuse to acknowledge something which you know to be real.
Admiral Horatio Nelson is supposed to have said this when wilfully disobeying a signal to withdraw during a naval engagement. Tales of that sort, especially when they are about national heroes like Nelson, tend to be exaggerated or entirely fictitious. That doesn't appear to be the case here though and there's very good evidence to show that Nelson was indeed the source of this phrase.
In the naval battle of Copenhagen in 1801 Nelson lead the attack of the British fleet against a joint Danish/Norwegian enemy. The British fleet of the day was commanded by Admiral Sir Hyde Parker. The two men disagreed over tactics and at one point Hyde Parker sent a signal (by the use of flags) for Nelson to disengage. Nelson was convinced he could win if he persisted and that's when he 'turned a blind eye'.
In their biography Life of Nelson, published just eight years later, Clarke and M'Arthur printed what they claimed to be a Nelson's actual words at the time:
[Putting the glass to his blind eye] "You know, Foley, I have only one eye - and I have a right to be blind sometimes... I really do not see the signal."
The first recorded use of the phrase in the form we normally use it today is in More letters from Martha Wilmot: impressions of Vienna, 1819-1829. These were reprinted in 1935 and this quotation is recorded as being sent by Ms. Wilmot in 1823:
"turn a blind eye and a deaf ear every now and then, and we get on marvellously well."
The manner of use of the phrase in that quotation suggests that it was well understood at the time.
See also: Kiss me Kardy