Certainly some of their funding comes from paid memberships. Beyond that, here is what my (not remotely extensive) research found:

In 2014 ProtonMail is listed as a Boston finalist for MassChallenge, (the same MassChallenge Boston who runs a US Air Force Lab), through whose programs start-ups

participate in a four-month, industry-agnostic accelerator where they receive:

• Hands-on support from top mentors and experts
• Free co-working space
• Access to MassChallenge’s unrivalled network of corporate partners
• Tailored workshops and office hours
• The opportunity to win a portion of more than $2M in cash prize

I am unsure if the MassChallenge application resulted in a grant, though there are finance sites that suggest it did.

Also in 2014, they raised $550,377 from backers on IndieGoGo In their campaign they said this:

We firmly believe that ProtonMail can only succeed in its mission if it remains independent. By raising money through crowd funding, we can ensure that our first and only priority is protecting the privacy of our users.

There are certain powerful governments and corporations out there who are in the business of controlling and exploiting personal data that will try to hinder us. If we want to live in a future where privacy on the internet is respected, we must stand together now and fight for those rights. With your support, we can make this brighter future a reality.

And yet...

They took 2 Million USD from venture capitalists at Charles River Ventures (CRV) and from FONGIT, described by ProtonMail as "a foundation supporting innovation on behalf of the State of Geneva and the Swiss Federal government". The head of ProtonMail's Advisory Board also serves as Director of FONGIT.

There have been people who claim to connect CRV to In-Q-Tel, the venture capital arm of the CIA, but my (very brief) research only came up with a loose connection through an employee. I did, however, find out an unrelated piece of information I would not have otherwise: In-Q-Tel lists GitLab in their portfolio. There is another (stronger) connection to the US government through CRV; a partner was appointed by Obama in 2012 to represent the US at the UN General Assembly.

It is also of note that Protonmail is advised by the MIT Venture Mentoring Service which is exclusively for the MIT community. I mention this because Protonmail has scrubbed much of the MIT influence, presumably because they think it will harm their business. This is the "about" now:

ProtonMail was founded in 2013 by scientists who met at CERN and were drawn together by a shared vision of a more secure and private Internet. Since then, ProtonMail has evolved into a global effort to protect civil liberties and build a more secure Internet, with team members also hailing from Caltech, Harvard, ETH Zurich and many other research institutions.

And this was that same page in 2014:

ProtonMail was founded in summer 2013 at CERN by scientists who were drawn together by a shared vision of a more secure and private Internet

ProtonMail is developed both at CERN and MIT and is headquartered in Geneva, Switzerland. We were semifinalists in 2014 MIT 100K startup launch competition and are advised by the MIT Venture Mentoring Service