Why is LAST_VALUE() not working in SQL Server?

Try adding ROWS BETWEEN UNBOUNDED PRECEDING AND UNBOUNDED FOLLOWING

I believe the default window for analytic functions that includes an ORDER BY ends at the current row.

LAST_VALUE(IsFullTime) OVER (
    PARTITION BY EntityId, EmployeeId
    ORDER BY EntityId, EmployeeId, PayPeriodStart
    ROWS BETWEEN UNBOUNDED PRECEDING AND UNBOUNDED FOLLOWING
) AS LastIsFullTimeValue

Here's the trick. I use FIRST_VALUE + DESC ordering (instead of LAST_VALUE + ASC ordering). Works. It is the shortest way and it let you avoid dreaded ROWS BETWEEN UNBOUNDED PRECEDING AND UNBOUNDED FOLLOWING.

FIRST_VALUE(IsFullTime) OVER (
    PARTITION BY EntityId, EmployeeId
    ORDER BY EntityId DESC, EmployeeId DESC, PayPeriodStart DESC
) AS LastIsFullTimeValue

Agreed, it is confusing why LAST_VALUE is not intuitive and requires so much attention compared to its twin sister FIRST_VALUE.

I believe you are looking to use ROW_NUMBER() and the get the last value based on payperiodstart date:

SELECT t.EntityId
    ,t.EmployeeId
    ,t.LastIsFullTimeValue
FROM (
    SELECT EntityId
        ,EmployeeId
        ,ROW_NUMBER() OVER (
            PARTITION BY EntityId
            ,EmployeeId ORDER BY PayPeriodStart DESC
            ) AS rn
        ,LastIsFullTimeValue
    FROM dbo.Payroll
    WHERE EmployeeId = 316691   -- you could probably put this in your outer query instead
        AND PayPeriodStart <= '12/31/2014'
        AND PayPeriodEnd >= '1/1/2014'
    ) t
WHERE t.rn = 1;