I would like to rent an apartment in Paris for a month, but I would like to have some type of concierge service, in case of emergency or a special need.
Any recommendations?
Suggest that you look for apart-hotels instead.
Guest Apartment Services. Their apartments are not for the budget minded if that matters to you. Haven in Paris (also not for the budget minded) also offers such services. You can read about what they offer at their websites.
Edit (1/9): Take a look at La Clef Louvre, which used to be the Citadines Suites Louvre. My brother-in-law and sister-in-law stayed there in 2014 and liked the combination of front desk services and private apartment. There are also hotels in Paris that have apartments where you can get the help of a concierge/front desk if you need it -- for example the Hotel Louison and Hotel Elysees Union.
Citadines Apart'hotels is a big international chain of apartment-hotels, providing desk service while avoiding the current debates about the legality of private apartment rentals. There are others. Big reservation services such as Booking.com and Hotels.com list them.
The best would be a 24-hour a day desk service - such as you would find in a hotel or an apart'hotel.
What is your budget and what time of year are you coming to Paris?
There are several apart'hotels, besides Citadines:
residencehenri4.com/en/ - in the 6th arrondissement
www.tripadvisor.com/Hotel_Review-g187147-d7377792-Reviews-Le_Roi_de_Sicile-Paris_Ile_de_France.html - in the 4th arrondissement