It is your own choice whether you want to have 'household contents insurance' or not. In case of theft, the usual things that are stolen are TVs, Laptops (of the tenants, which are not insured under your policy) or other such electronic gadgets.
If you end up getting such insurance, please remember to inform your insurance company that you are renting out your furnished flat and that it is additionally insured against wanton negligence.
The quality of an insurance policy or company only becomes clear once an accident happens. Basically, it depends on whether the company is willing to pay or not - and in most cases they tend not to. It is better to properly secure the apartment rather than going through the drawn-out process with the insurance company and discussions with the tenant, should an accident or theft indeed occur. Most important is to secure the doors and windows. On average, a burglar tries for around 5 minutes to break into an apartment. If he or she cannot break in to the apartment in less than 5 minutes, the burglar will tend to gives up and move on to the next one. Of course having more security doesn’t mean that there will be no break in at all. It just means that the chances of a burglary are very low. You should also look at the security instructions of the police in your area.
Please make sure that there are no copies made of the keys that you have provided originally to the tenant. Having a security lock makes sure that only you are authorised to make a copy of the keys.
Make sure to check whether all the doors and windows are secured properly against theft – especially if your apartment is on the ground floor, the loft or on the mezzanine. Loft apartments are usually in danger because the burglar does not have to fear being interrupted by other residents of the building walking by. Also make sure to check the security locks on the ceiling windows of loft apartments and also inform your tenant to do the same when not in the apartment. For ground floor apartments, one should always be very careful. You should make sure that the windows cannot be opened fully when left in a tilted state.
Tenants should, for obvious reasons, not smoke in the apartments they are renting. But still you should install a smoke detector. Fires usually do not occur in entire apartments, but in the last years there have been a series of blazes in rented apartments. A smoke detector is available for less than 10 Euros.