Why yes, Romance languages do have vowel harmony, or what the Romance linguistics literature likes to call **metaphony**. The following information on Spanish is all from [Alkire & Rosen 2010][1], which is my only source of knowledge about Spanish phonology.

Alkire & Rosen 2010 devote subchapter 5.2 to "yod effects in Spanish". The effect of raising of a stressed vowel under the influence of a following high vowel gesture (anticipation) seems to be especially prominent in the verb system, where it has been largely morphologized. But you find it in nouns as well, so you have:

> VĬTREU   'glassy'   vidrio   'glass'
> 
> SĒPIA   'cuttlefish'   jibia
> 
> PLŬVIA   'rain'   lluvia

The stressed vowels in Latin should have given Romance high mids, which in Spanish should go on to give you mids: [e] and [o]. Insead, what you you have is [i] and [u], evidence for anticipatory raising. Alkire & Rosen give many more examples.

Yod effects can also be found in **Italian**, but they're more restricted.

> PŬGNU   'fist'   pugno
> 
> FAMILĬA   'family'   famiglia
> 
> CONSĬLIU   'counsel'   consiglio

Same story here as for Spanish, except it happens before [ɲ] and [ʎ]. There are other examples as well, including examples where the expected raising fails to happen.

**French** has a lot of yod-related effects too. See ch. 5.3 of Alikre & Rosen 2010.

Maybe someone else can fill us in on Portuguese. :)

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**Romanian** has this stuff too, but it involves stressed low vowels alternating with mids before an [j] in the inflection (a-ə, e̯a-e, o̯a-o; note that the diphthongs are considered phonemic) and before an [e] in the plural marker (only e̯a-e). This correlates with gender in nouns and person in verbs, so it's also morphologized to a large extent. See Chitoran 2002 ([abstract][2]).

Nouns:
> [mare] &emsp; 'sea' &emsp; [mər<sup>j</sup>] &emsp; 'seas'
> 
> [se̯arə] &emsp; 'evening' &emsp; [ser<sup>j</sup>] &emsp; 'evenings'
> 
> [bo̯alə] &emsp; 'disease' &emsp; [bol<sup>j</sup>] &emsp; 'diseases'
 
Verbs:
> [aratə] &emsp; 'show.3rd.sg' &emsp; [arəʦ<sup>j</sup>] &emsp; 'show.2nd.sg'
>
> [vise̯azə] &emsp; 'dream.3rd.sg' &emsp; [visez<sup>j</sup>] &emsp;
> 'dream.2nd.sg'
> 
> [do̯arme] &emsp; 'sleep.3rd.sg' &emsp; [dorm<sup>j</sup>] &emsp;
> 'sleep.2nd.sg'


  [1]: http://www.amazon.com/Romance-Languages-A-Historical-Introduction/dp/0521717841/
  [2]: http://www.dartmouth.edu/~linguist/pdfs/probus.pdf