If it were my car, I'd replace the tensioner spring too. You should be ok with starting and running the car with the old spring. But at some point my recommendation is to add the spring into your next parts order and replace it.

Not as critical on the turbo engines as on the 240D (the 240D tensioner has no ratchet mechanism, at least not in 1981 models) but still you don't want a broken spring.

Ken300D

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1982 300D at 351K miles
1984 300SD at 217K miles
1987 300D at 370K miles