"Do" replacing "Please"

Solution 1:

The usage is different, and dependent upon situation and social station. For example, the first usage "Do let us know your thoughts", indicates that the speaker is trying to make an impression of honest interest rather than mouthing a polite but meaningless offer. Again, "Do let me know." would indicate that the speaker is genuine, while "Please let me know." is ambiguous - the speaker may actually want to know, or may only be mouthing social niceties.

I would say that "Do consider the proposal." is a different usage, also imperative, but commanding rather than making a request. It would be used by a superior speaking to a subordinate while giving direction. Someone speaking to a person of equal or higher status would come off as rude, and should instead use "Please consider the proposal".

Solution 2:

In my experience, versions starting with Do rather than Please are more polite. To my ear, both versions sound ok, and which is used will be more a matter of what one is used to using than to any other factor. I would expect to find disagreement about which version is thought to be more polite. Also note that ngrams for phrases with tell and ngrams for phrases with let suggest that versions with Do were more common before about 1940, and versions with Please more common after that.

Solution 3:

The tone in which the two words are used in a sentence is different. I think that 'do call me' emphasizes on the hold/impact of the person saying it. However, 'please call me' sounds weak. And therefore, the tone in which these both sentences are used is quite different.