As a native English speaker (American) I don't think it is right to say spare in this sense. Because spare is mostly synonymous with save but not completely. To save something can be used in this context, but sparing is meant as saving from damage, death, or some other outcome. So as someone pointed out "Spare me the details" uses the word spare in that way, but it is an expression that has maintained that form for a long time, but the common usage has changed too much to use it in that way outside of certain phrases. (All of this might be pretty insubstantial, but language is fun.)
I agree, the connotations of spare don't really fit with an inconsequential object such as a newspaper. You /could/ say "I have a spare newspaper", or "I will spare the newspaper FOR the fire", but it would sound rather odd to say "I will spare the newspaper FROM the fire". If the newspaper was an artifact, perhaps... Just to be sure, I checked the dictionary to make sure there wasn't some obscure usage of the word that would apply here, but it doesn't seem so: http://dictionary.reference.com/browse/spare It's a good way of remembering the Swedish word though!