No, chinchillas should not be eating tissues at all.
They can however chew on the tissues but should not be swallowing them as they cannot be digested.
However chins do love to chew in tissues. Make sure that they do not swallow them though as they are bad for their stomach.
Just give tissues to your chinchilla to play with or for their bedding only.
Be careful and watch them though. If they start to actually eat the tissues you should avoid giving tissues to your chin as it will cause health issues.