Well, I understand the question to be in two parts.
(1) Whether the actual pitch of Old Japanese verbs is known (to researchers);
(2) How do modern Japanese people apply modern pitch when reading Old or Classical texts.
These are properly two different questions, so I will try to deal with them separately.
(1) The historical pitch is really well-known from the tone marks assigned to many words in dictionaries and texts. The go-to reference here is Elisabeth de Boer's The Historical Development of Japanese Tone (2 Vols., Harrassowitz Verlag, 2010).
Usually, in literature dedicated to Japanese tones only nouns are considered, but the tonal marks occurred on verbs as well, and there are listings of those.
For example, in Samuel E. Martin The Japanese Language Through Time (Yale University Press, 1987), which is the most accessible place actually giving historical tones, on pages 192-197 there is the complete chart of all verbs marked with tonal marks in the Ruiju Myōgishō dictionary (12th cent.), in several forms. (Note that Martin assumed the tonal marks to denote the opposite of how they are treated now, so read all, say 'HL' as 'LH' and vice versa.) For example, from Ruiju Myōgishō we know that the verb 置く 'to put' has final form [おく]{LL}, coordinate converb [おきて]{LHL} without onbin or [おいて]{LHL}, [おいて]{LHH} with one, infinitive (converb) [おき]{LH} and deverbal noun [おき]{LL}.
Of course, all of the aforementioned refers to the period of tone marks, which is late 10th to 14th century, corresponding to Early Middle Japanese; however, it is not assumed that Old Japanese of the 萬葉集 was drastically different in this regard.
(2) The pronunciation of Classical verbs in modern, Tokyo, pronunciation is a separate matter, more prescriptive than descriptive. For this, there are norms, and dictionaries actually explain that.
アクセント習得法則, the preface to the Shinmeikai accent dictionary, actually has Table 2 of Section 42 to help making the Tokyo pitch for a Classical correspondence to a current verb. In general, most forms are made as in current language, as for the unique forms:
The final (終止形) and attributive (連体形) forms are unstressed or stressed according to the verb (泣く is unstressed in modern, so [なく]{LH}, [なくもの ]{LHHHL}; 起きる is stressed in modern, so [おく]{HL}, [おくるもの]{LHLLL}). Post-final らん is stressed on ら if the verb is unstressed, allows the stress on the verb otherwise ([なくらん]{LHHL}, but [おくらん]{HLLL}).
The a-form (未然形) is as follows: negation ず does not enforce any stress ([なかず]{LHH}, [おきず]{HLL}), same with its attributive -ぬ; conjectural -ん (-む) enforces the stress before it, overriding all else ([なかん]{LHL}, [おきん]{LHL}); conditional -(a)ば makes stress before it if unstressed ([なかば]{LHL}) but gives way to existing stress ([おきば]{HLL}).
Coordinate converb -て enforces stress before it only on unstressed ones ([なきて]{LHL}, but [おきて]{HLL}; the dictionary also says there is a tendency to say the -て-form of unstressed verbs unstressed as [なきて]{LHH}); perfective -ぬ enforces stress before ([なきぬ]{LHL}, [おきぬ]{LHL}); -たり enforces stress before it only on unstressed ones ([なきたり]{LHLL}, but [おきたり]{HLLL}) - here also [なきたり]{LHHL} is possible, with stress on た in case of unstressed verbs.
As for the 已然形 form, -ども enforces stress before it only on unstressed ones ([なけども]{LHLL}, but [おきども]{HLLL}).
Finally, in the imperative, the unstressed verbs get final stress (or before -よ, if present: [なけ ]{LHL}, [なけよ]{LHL}), the others retain stress ([おき]{HL}, [おきよ]{HLL}).
