I currently have a very large list of user emails (around 100K), but with unverified emails addresses. In order to make signing up and using the site as convenient as possible (especially for the mobile apps), I did not require email verification.
I am running a last-effort fund-raiser for my site, and I would like to contact those users (because they would be the most interested in helping with the fund-raiser). Is there an optimal way to do this?
Well, this is risky but the best thing you can do is make yourself look as un-spammy as possible. Following the Can Spam Act guidelines is a good place to start:
A visible and operable unsubscribe mechanism is present in all emails.
Consumer opt-out requests are honored within 10 business days. 
Opt-out lists also known as Suppression lists are only used for compliance purposes.
Accurate from lines (including "friendly froms")
Relevant subject lines (relative to offer in body content and not deceptive)
A legitimate physical address of the publisher and/or advertiser is present. PO Box addresses are acceptable in compliance with 16 C.F.R.
316.2(p) and if the email is sent by a third party, the legitimate physical address of the entity, whose products or services are
promoted through the email should be visible.
A label is present if the content is adult.
Sending behavior compliance
A message cannot be sent through an open relay
A message cannot be sent without an unsubscribe option.
A message cannot be sent to a harvested email address
A message cannot contain a false header
A message should contain at least one sentence.
A message cannot be null.
Unsubscribe option should be below the message.
Using a third party service like Constant Contact will take care of much of this for you (after an initial configuration). They also won't do things like send it all from the same server and do other good technical things that help prevent mail from being considered spam.