Are immigration lawyers forced to decline clients because the prospective new client is unable to pay the immigration lawyer their fee?

Let's first of all position who are the audiences for an Immigration Firm. If we classified immigration into:

1- Skilled immigration (Employees), and
2- Business Immigration (Investors)

Skilled Immigration fees are affordable and due to the high competition the Professional Fees are competitive and payment plans are there to attract more clients.
HINT: Despite that, Applicants have to conduct an overall check up about the credibility of their meant immigration consultants.

On the other hand, Business Immigration which involves a bigger amounts of fees and investment are treated somehow different.
As per the country of destination and the route of immigration which is almost limited between Property Investment, Financial Instruments Investment and Donation varieties of payments plans are offered to attract Investors.
In other words, Legal Fees and Processing Fees are considered as a small portion of the Deal which doesn't shape a concern to Migrant Investors rather than the tool used for immigration and the success rate of that firm.
HINT: When choosing an immigration program it is preferable not to pick Immigration firms relying only on RCBI programs (Residency & Citizenship by Investment) as the risk lies in the project continuity as per the demand which to some extent can delay the whole immigration files under that project.

No.
Absolutely not.
But they should.
If they want to stay in business and be able to continue to service future immigration clients who are able to pay.

In Canada, there are certain cases that allow Legal Aid in Ontario. We also have other immigration professionals like Regulated Canadian Immigration Consultants that may work at a reduced rate. I also know professionals that offer payment plans. I have not heard of other lenders in this sphere aside from people using credit cards.