Origin of phrasal verb "love on"
Solution 1:
From John Dryden's 1700 Palamon and Arcite (a translation of The Knight's Tale from Chaucer's Canterbury Tales)...
So thou, if fortune will thy suit advance,
Love on, nor envy me my equal chance
Two guys both fancy the same girl; less poetically, the speaker is saying they both have the right to try their luck and love on (i.e. - continue to "love").
My guess is Dryden's love = pursue your suit (archaically, make love to her). But you could just as easily interpret it as experience feelings of love for her. The point is there have always been contexts where (verb) love can be followed by on.
But I don't think there will be any "acceptable" instances of (verb) love on [object of love], in the sense of being considered "grammatical" by any significant proportion of Anglophones.
OP's two instances aren't really the same. The first [mistakenly] substitutes on for towards (AmE toward), probably echoing preceding in the streets. The second is just a quirky "slogan" format.
Solution 2:
It's a slang term, which loosely means to shower someone (or something) with attention. A group of three sisters that I rode horses with when I was very young used the term quite often, as did every member of their family. For example, if a horse was being especially biddable, they would say "He done good! Be sure and love on 'im!", or "I just sat on 'im for a while and rubbed and loved on 'im." I'm not absolutely positive where the girls were from originally, but I believe they mentioned Alabama or Arkansas. This was in Northwest Oregon that I was acquainted with them, during the 70's and early 80's.
At the risk of sounding ridiculously judgmental, the phrase, for me, has a bit of a backwoods, low rent, and/or "white trash" (for lack of a better description) connotation. It has been many years now since I last spoke with any of the girls or their family, so I can't say for certain if those undertones come from the fact that the only times that I've ever heard it used, it was usually by the poorest of folk back then, or if it stems from my memory of those girls and their family in particular. Perhaps the phrase originates in the South, where the family itself had originated from.
Solution 3:
I agree with professor_feather -- I have definitely seen this in the Christian community. Based on the content, I expect your examples are influenced by this -- probably from a church bulletin or Christian non-profit.
The Dictionary of Christianese gives a definition that matches my experience:
http://www.dictionaryofchristianese.com/love-on/
love on v. phr. Sometimes also just love on. To demonstrate care, concern, or platonic affection for someone by using words, actions, or non-sexual physical touch.
This source tracks usage in Christian books as early as 1993, though it was probably in use before then.
1993 Mize The Church Without Spot or Wrinkle 80 : Treating them like royalty is only a small part of what God has planned for you when you invite them over for dinner. It’s a chance for you to love on them. Did you know that agape love never fails? Love on them like you would Jesus.
Note the reference to the Greek word agape. Put simply, Greek has several words which are translated as love. This sometimes has bad results when translating from Greek to English:
Consider John 21:15-17
15 When they had finished eating, Jesus said to Simon Peter, "Simon son of John, do you love (agape) me more than these?" "Yes, Lord," he said, "you know that I love (philia) you." Jesus said, "Feed my lambs." 16 Again Jesus said, "Simon son of John, do you love (agape) me?" He answered, "Yes, Lord, you know that I love (philia) you." Jesus said, "Take care of my sheep." 17 The third time he said to him, "Simon son of John, do you love (philia) me?" Peter was hurt because Jesus asked him the third time, "Do you love (philia) me?" He said, "Lord, you know all things; you know that I love (philia) you." Jesus said, "Feed my sheep.
I've marked the different words that are translated as "love". The English version seems to imply that Jesus is asking the same question, even after getting the answer, but the Greek shows that Peter didn't really answer the first two questions -- he used a different word.
While it wouldn't make sense to change the translation to "*Simon son of John, do you love on me?", Christians use "love on" to refer to that idea of agape love. While it might not exactly match the Greek definition, agape is often described in Christian messages as "love, the verb." It contrasts with the intellectual philia and primal eros -- it is love displayed through actions. zeromus wrote this in the comments, which is a good summary, I think:
You literally can't love on the homeless without working in the streets (but you can love them from a sanitary distance by writing someone a check).
Those were reasons why the Christian community embraced the word, but I think Anita Valium is correct about the etymology. Commenters in the Dictionary of Christianese also agree that the phrase is Southern in origin. The South also has a large Christian community, so it has spread from there.