T-SQL's DATEADD is not a "real" function , it's actually a language-feature (like a statement or control-structure) and the first parameter ( datepart ) accepts keyword arguments, not value arguments, so you cannot parameterize it using a @variable nor column expression...

...instead, you'll need to use a CASE expression:

SELECT
    t.*,

    CASE t.Interval
        WHEN 'year'  THEN DATEADD( year , t.Increment, t.Latest_Appointment_Date )
        WHEN 'month' THEN DATEADD( month, t.Increment, t.Latest_Appointment_Date )
        WHEN 'day'   THEN DATEADD( day  , t.Increment, t.Latest_Appointment_Date )
        WHEN 'week'  THEN DATEADD( week , t.Increment, t.Latest_Appointment_Date )
        WHEN 'hour'  THEN DATEADD( hour , t.Increment, t.Latest_Appointment_Date )
        /* etc for the other defined keywords, tedious, huh? */
    END AS NextDate
FROM
    #Dummy AS t

My answer is basically what Dai said. Here's an inline table valued function you can use to abstract away that logic so you can reuse it elsewhere if you need (I copied all the various possible inputs to the proc from the DATEADD documentation)

create or alter function dbo.DynamicDateAdd
(
    @Interval varchar(30),
    @Increment int,
    @Date date
)
returns table
as
return
select
    NewDate = case
        when @Interval in ('year', 'yy', 'yyyy') then dateadd(year, @Increment, @Date)
        when @Interval in ('quarter', 'qq', 'q') then dateadd(quarter, @Increment, @Date)
        when @Interval in ('month', 'mm', 'm') then dateadd(month, @Increment, @Date)
        when @Interval in ('dayofyear', 'dy', 'y') then dateadd(dayofyear, @Increment, @Date)
        when @Interval in ('day', 'dd', 'd') then dateadd(day, @Increment, @Date)
        when @Interval in ('week', 'wk', 'ww') then dateadd(week, @Increment, @Date)
        when @Interval in ('weekday', 'dw', 'w') then dateadd(weekday, @Increment, @Date)
        when @Interval in ('hour','hh') then dateadd(hour, @Increment, @Date)
        when @Interval in ('minute','mi', 'n') then dateadd(minute, @Increment, @Date)
        when @Interval in ('second', 'ss', 's') then dateadd(second, @Increment, @Date)
        when @Interval in ('millisecond', 'ms') then dateadd(millisecond, @Increment, @Date)
        when @Interval in ('microsecond', 'mcs') then dateadd(microsecond, @Increment, @Date)
        when @Interval in ('nanosecond', 'ns') then dateadd(nanosecond, @Increment, @Date)
        else null
    end

go

Example Usage

declare @Now date = getdate()

;with inputs (Interval, Increment) as
(
    select 'day', checksum(newid()) % 3 union all
    select 'week', checksum(newid()) % 3 union all
    select 'month', checksum(newid()) % 3 union all
    select 'year', checksum(newid()) % 3 union all
    select 'not-an-input', checksum(newid()) % 3 
)
select
    Now = @Now,
    a.Increment,
    a.Interval,
    b.NewDate
from inputs a
cross apply dbo.DynamicDateAdd(interval, Increment, @Now) b