The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\, v. i.
   To require; to demand. [Obs.]
   [1913 Webster]

         We shall write to you,
         As time and our concernings shall importune. --Shak.
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The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\ ([i^]m`p[o^]r*t[=u]n"), a. [F.
   importun, L. importunus; pref. im- not + a derivative from
   the root of portus harbor, importunus therefore orig.
   meaning, hard of access. See Port harbor, and cf.
   Importunate.]
   1. Inopportune; unseasonable. [Obs.]
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   2. Troublesome; vexatious; persistent; urgent; hence,
      vexatious on account of untimely urgency or pertinacious
      solicitation. [Obs.]
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            And their importune fates all satisfied. --Spenser.
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            Of all other affections it [envy] is the most
            importune and continual.              --Bacon.
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The Collaborative International Dictionary of English v.0.48:

Importune \Im`por*tune"\, v. t. [imp. & p. p. Importuned
   ([i^]m`p[o^]r*t[=u]nd"); p. pr. & vb. n. Importuning.]
   [From Importune, a.: cf. F. importuner.]
   1. To request or solicit, with urgency; to press with
      frequent, unreasonable, or troublesome application or
      pertinacity; hence, to tease; to irritate; to worry.
      [1913 Webster]

            Their ministers and residents here have perpetually
            importuned the court with unreasonable demands.
                                                  --Swift.
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   2. To import; to signify. [Obs.] "It importunes death."
      --Spenser.
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WordNet (r) 3.0 (2006):

importune
    v 1: beg persistently and urgently; "I importune you to help
         them" [syn: importune, insist]