As @Pavel answered, the code in the question does not make a whole lot of sense.
Either way, according to your description, you do not need a function at all.

I want to create a function that takes each of those years and runs a query on another table to count the number of rows that belonging to that year.

"Belonging to a year" seems to mean overlapping time range. Use a simple query with a JOIN, a smart join condition and aggregation.

Assuming this table (missing in the question):

CREATE TABLE years (
  the_date date PRIMARY KEY
)

This query would do the job:

SELECT the_date, count(*) AS row_ct
FROM   years y
JOIN   other_table o ON (y.the_date, y.the_date + interval '1 year')
               OVERLAPS (o.valid_from, COALESCE(o.valid_to, 'infinity'))
GROUP  BY 1;

Details for the OVERLAPS operator:
Find overlapping date ranges in PostgreSQL

The manual about 'infinity'.

What @Pavel said. Plus, according to your description, you do not need a function at all.

I want to create a function that takes each of those years and runs a query on another table to count the number of rows that belonging to that year.

"Belonging to a year" seems to mean "overlapping time range". Use a simple query with a JOIN, a smart join condition and aggregation.

Assuming this table (missing in the question):

CREATE TABLE years (
  the_date date PRIMARY KEY
);

This query would do the job:

SELECT the_date, count(*) AS row_ct
FROM   years y
JOIN   other_table o ON (y.the_date, y.the_date + interval '1 year')
               OVERLAPS (o.valid_from, COALESCE(o.valid_to, 'infinity'))
GROUP  BY 1;

About the OVERLAPS operator:

• Find overlapping date ranges in PostgreSQL

The manual about 'infinity'.

As @Pavel answered, the code in the question does not make a whole lot of sense.
Either way, according to your description, you do not need a function at all.

I want to create a function that takes each of those years and runs a query on another table to count the number of rows that belonging to that year.

"Belonging to a year" seems to mean overlapping time range. Use a simple query with a JOIN, a smart join condition and aggregation.

Assuming this table (missing in the question):

CREATE TABLE years (
  the_date date PRIMARY KEY
)

This query would do the job:

SELECT the_date, count(*) AS row_ct
FROM   years y
JOIN   other_table o ON (y.the_date, y.the_date + interval '1 year')
               OVERLAPS (o.valid_from, COALESCE(o.valid_to, 'infinity'))
GROUP  BY 1;

Details for the OVERLAPS operator:
Find overlapping date ranges in PostgreSQL

The manual about 'infinity'.

As @Pavel answered, the code in the question does not make a whole lot of sense.
Either way, according to your description, you do not need a function at all.

I want to create a function that takes each of those years and runs a query on another table to count the number of rows that belonging to that year.

"Belonging to a year" seems to mean overlapping time range. Use a simple query with a JOIN, a smart join condition and aggregation.

Assuming this table (missing in the question):

CREATE TABLE years (
  the_date date PRIMARY KEY
)

This query would do the job:

SELECT the_date, count(*) AS row_ct
FROM   years y
JOIN   other_table o ON (y.the_date, y.the_date + interval '1 year')
               OVERLAPS (o.valid_from, COALESCE(o.valid_to, 'infinity'))
GROUP  BY 1;

Details for the OVERLAPS operator:
Find overlapping date ranges in PostgreSQL

The manual about 'infinity'.